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ELEMENTARY  TREATISE 


ON 


MECHANICS, 


FOR  THE  USE   OP 


COLLEGES  AND   SCHOOLS   OF   SCIENCE. 


BY 


WILLIAM  G.  PECK,  Ph.D..  LL.D., 

PROPFSSOR     OF     MATHEMATICS    AND    ASTRONOMY    IN     COLUMBIA    COLLEGE,    AND    OJ" 
MBCHANIC8    IN    THE    SCHOOL    OF    MINB8. 


NEW  YORK    :•  CINCINNATI  •:.  CHICAGO 

AMERICAN     BOOK      COMPANY 


PROM  THK  PRBSS  OP 

A.    S.     BARNES    &    CO. 


•^v^-i 


r3c 


i>L.lered  eccordlng  to  Act  of  Congress,  in  the  year  1870,  0/ 

WILLIAM  G.  PECE , 

In  tha  Office  o<   tke  Librarian  of  CougreiMi,  at  Washinxtofi 


GIFT  OF 


PEEFAOE. 


The  following  Treatise  was  originally  prepared  to  sup- 
ply a  want  felt  by  the  compiler,  whilst  engaged  in  teaching 
Natural  Philosophy  to  college  classes.  It  is  now  proposed 
to  introduce  into  it,  in  a  simplified  form,  the  results  of 
many  years'  experience  in  its  use  as  a  text-book.  To  ac- 
complish this,  the  entire  book  has  been  rewritten,  the  de- 
scriptive matter  condensed,  the  demonstrations  simplified, 
and  the  practical  scope  of  the  work  extended ;  but  in  no 
instance  has  any  essential  principle  been  omitted.  The 
most  important,  if  not  the  only,  change  in  the  plan  of  the 
work  is  the  omission  of  the  Calculus.  This  change  has 
been  made,  to  cause  the  work  to  conform  more  closely 
to  the  original  design,  which  was,  to  produce  a  book 
that  should  form  a  suitable  connecting  link,  between 
purely  popular  works,  on  the  one  hand,  and  those  of  the 
highest  grade,  on  the  other.  In  most  of  our  Colleges, 
the  Calculus  is  either  not  taught  at  all,  or  else  its  study 
is  made  optional,  and  pursued  without  reference  to  its 
use  as  a  tool  for  scientific  investigation.  The  change 
referred  to,  brings  this  edition  of  the  work  within  the 
range  of  the  College  Curriculum,  and  it  is  hoped  does 


4  PKEFACE. 

not  impair  its  value  as  a  text-book  for  Schools  of  Science. 
As  modified,  it  embraces  all  the  elementary  propositions 
of  Mechanics,  arranged  in  logical  order,  rigidly  demon- 
strated and-  fully  illustrated  by  practical  examples;  its 
scope,  sufficiently  extended  to  meet  the  wants  of  Colleges 
and  Schools  of  Science;  its  treatment,  so  simple  that  it 
may  be  read  with  profit  by  those  who  have  not  the 
leisure  to  make  the  mathematical  sciences  a  specialty;  and 
its  plan,  such  as  to  render  it  a  suitable  introduction  to 
those  higher  treatises  on  Mechanical  Philosophy,  that  all 
must  read  who  would  appreciate  and  keep  pace  with  the 
discoveries  of  modern  science. 

Columbia  College, 
June  ITth,  1870. 


COI^TEE-TS 


CHAPTER  I. 

DEFINITIONS  AND   INTRODUCTORY   REMARKS. 
Art.  PAGB 

1.  Definition  of  a  Body 11 

2.  Rest  and  Motion  H 

3.  Rectilinear  and  Curvilinear  Motion U 

4.  Motion  of  Translation  and  Rotation 12 

5.  Uniform  and  Varied  Motion 12 

6.  Definition  of  a  Force 12 

7.  Classification  of  Forces 12 

8.  Extraneous  Forces 13 

9.  Molecular  Forces 13 

10.  Constitution  and  Classification  of  Bodies 13 

11.  Essential  Properties  of  Bodies 14 

12.  Laws  of  Motion 15 

13.  Secondary  Properties  of  Bodies 15 

14.  Force  of  Gravity  and  Weight 16 

15.  Mass  and  Density 17 

16.  Momentum,  or  Quantity  of  Motion 18 

17.  Measure  of  Forces 19 

18.  Acceleration  due  to  a  Force 20 

19.  Representation  of  Forces 20 

20.  Equilibrium 21 

21.  Definition  of  Mechanics 21 

CHAPTER  n. 

COMPOSITION,  RESOLUTION,  AND  EQUILIBRIUM  OP  FORCES. 

22.  Definitions 23 

23.  Composition  of  Forces  whose  Directions  coincide 23 

24.  Composition  of  Concurrent  Forces 24 

25.  Parallelogram  of  Forces 25 

26.  Geometrical  Applications  of  the  Parallelogram  of  Forces. . .  26 

27.  Polygon  of  Forces 27 

28.  Parallelopipedon  of  Forces 28 

29.  Components  of  a  Force  in  the  Direction  of  Rectangular  Axes  28 

30.  Analytical  Composition  of  Rectangular  Forces 30 

51.  Application  to  Groups  of  Concurrent  Forces 32 


6  CONTENTS. 

Art.  PA6« 

33.  Formula  for  the  Resultant  of  two  Forces •  •  34 

33.  Relation  between  two  Forces  and  their  Resultant 35 

34.  Principle  of  Moments 36 

35.  Moment  of  a  Force  with  respect  to  an  Axis 38 

36.  Pi-inciple  of  Virtual  Moments 39 

87.  Resultant  of  Parallel  Forces 41 

38.  Geometrical  Composition  of  Parallel  Forces 44 

39.  Co-ordinates  of  the  Centre  of  Parallel  Forces 46 

40.  Composition  of  Forces  in  Space,  applied  at  points  invo-r'ably 

connected 47 

41.  Conditions  of  Equilibrium , 49 

CHAPTER  III. 

CENTRE   OP  GRAVITY  AND    STABILITt. 

42.  Weight 50 

43.  Centre  of  Gravity 50 

44.  Preliminary  Principles 51 

45.  Centre  of  Gravity  of  a  Straight  Line 51 

46.  Additional  Principles 52 

47.  Centre  of  Gravity  of  a  Triangle 53 

48.  Centre  of  Gravity  of  a  Parallelogram 53 

49.  Centre  of  Gravity  of  a  Trapezoid 54 

50.  Centre  of  Gravity  of  a  Polygon .  54 

51.  Centre  of  Gravity  of  a  Pyramid 56 

52.  Centre  of  Gravity  of  a  Prism 57 

53.  Centre  of  Gravity  of  a  Polyhedron 58 

54.  Experimental  Determination  of  the  Centre  of  Gravity 59 

55.  Centre  of  Gravity  of  a  System  of  Bodies 60 

56.  Stable,  Unstable,  and  Indifferent  Equilibrium 63 

57.  Stability  of  Bodies  on  a  Horizontal  Plane 65 

58.  Pressure  of  one  Body  on  another 67 

CHAPTER  IV. 

ELEMENTARY  MACHINES. 

59.  Definitions  and  General  Principles 74 

60.  Work 74 

61.  Trains  of  Mechanism 76 

62.  The  Mechanical  Powers 77 

63.  The  Cord 77 

64.  The  Lever 78 

65.  The  Compound  Lever 81 

66.  The  Elbow-joint  Press 82 

67.  Weighing  Machines 83 


coli^TE:^fTS.  7 

Art.  PAOB 

68.  The  Common  Balance 83 

69.  The  Steelyard 85 

70.  The  Bent  Lever  Balance 86 

71.  Compound  Balances 87 

73.  The  Inclined  Plane 89 

73.  The  Pulley 92 

74.  Single  Fixed  Pulley 92 

75.  Single  Movable  Pulley 93 

76.  Combination  of  Movable  Pulleys 94 

77.  Combinations  of  Pulleys  in  Blocks 95 

78.  The  Wheel  and  Axle 96 

79.  Combinations  of  Wheels  and  Axles 97 

80.  The  Crank  and  Axle,  or  Windlass 98 

81.  The  Capstan 99 

82.  The  Differential  Wmdlass 99 

83.  Wheel-work 101 

84.  The  Screw 103 

85.  The  Differential  Screw 105 

86.  The  Endless  Screw 106 

87.  The  Wedge 107 

88.  Application  of  the  Principle  of  Virtual  Moments 108 

89.  Hurtful  Resistances 109 

90.  Friction 109 

91.  Methods  of  Finding  the  CoeflScient  of  Friction 110 

92.  Influence  of  Friction  on  an  Inclined  Plane 112 

93.  Limiting  Angle  of  Resistance 113 

94.  Friction  on  an  Axle 115 

95.  Line  of  Least  Traction 115 

96.  Resistance  to  Rolling 116 

97.  Work  of  Friction 117 

98.  Adhesion 117 

99.  Stiffness  of  Cords 118 

100.  Atmospheric  Resistance 118 

CHAPTER  V. 

EECTILmEAR  AND   PERIODIC   MOTION. 

101.  Motion 119 

102.  Uniform  Motion 120 

103.  Uniformly  Varied  Motion 120 

104.  Application  to  Falling  Bodies 121 

105.  Motion  of  Bodies  projected  vertically  upward 124 

106.  Restrained  Vertical  Motion 126 

107.  Atwood's  Machine 127 


8  COKTENTS. 

Art.  PAGB 

108.  Motion  of  Bodies  on  Inclined  Planes 129 

109.  Motion  of  a  Body  down  a  succession  of  Inclined  Planes.. .  133 

110.  Periodic  Motion ,134 

111.  Angular  Velocity  and  Angular  Acceleration 135 

112.  The  Simple  Pendulum  136 

113.  De  r  Ambert's  Principle 139 

114.  The  Compound  Pendulum 140 

115.  Angular  Acceleration  of  a  Compound  Pendulum 141 

116.  Length  of  an  Equivalent  Simple  Pendulum 143 

117.  Reciprocity  of  Axes  of  Suspension  and  Oscillation 144 

118.  Practical  Application  of  the  Pendulum 147 

119.  Graham's  Mercurial  Pendulum 147 

120.  Harrison's  Gridiron  Pendulum 148 

121.  Basis  of  a  System  of  Weights  and  Measures 149 

122.  Centre  of  Percussion 151 

123.  Moment  of  Inertia 152 

124.  Centre  and  Radius  of  Gyration 155 

CHAPTER  VI. 

CURVILINEAR   AND    ROTARY   MOTION. 

125.  Motion  of  Projectiles 157 

126.  Centripetal  and  Centrifugal  Forces 164 

127.  Measure  of  the  Centrifugal  Force .  164 

128.  Centrifugal  Force  at  points  of  the  Earth's  Surface 167 

129.  Centrifugal  Force  of  Extended  Masses 170 

130.  Principal  Axes 172 

131.  Experimental  Illustrations.  . .    173 

132.  Elevation  of  the  Outer  Rail  of  a  Curved  Track 175 

133.  The  Conical  Pendulum 177 

134.  The  Governor 178 

135.  Definition  and  Measure  of  Work 181 

136.  Work  when  the  Power  acts  obliquely 182 

137.  Rotation 184 

138.  Quantity  of  Work  of  a  Force  producing  Rotation 184 

139.  Accumulation  of  Work 186 

140.  Living  Force  of  Revolving  Bodies 188 

141.  Fly-wheels 189 

442.  Composition  of  Rotations , 190 

143.  Application  to  the  Gyroscope 193 

CHAPTER  VIL 

<  MECHANICS   OF    LIQUIDS. 

144.  Classification  of  Fluids 196 


CONTENTS.  9 

Art.  PAOB 

145.  Principle  of  Equal  Pressures 196 

146.  Pressure  due  to  Weight 198 

147.  Centre  of  Pressure  on  a  plain  Surface 202 

148.  Buoyant  eflFort  of  Fluids 206 

149.  Floating  Bodies 207 

150.  Specific  Gravity 209 

151.  Metliods  of  finding  Specific  Gravity 211 

152.  Hydrostatic  Balance 211 

153.  Specific  Gravity  of  an  Insoluble  Body 211 

154.  Specific  Gravity  of  a  Soluble  Body 212 

155.  Specific  Gravity  of  Liquids 213 

156.  Specific  Gravity  of  Air 214 

157.  Hydrometers 215 

158.  Nicholson's  Hydrometer 215 

159.  Scale  Areometer 216 

160.  Volumeter 217 

161.  Densimeter 218 

162.  Centesimal  Alcoometer  of  Gay  Lussac 219 

163.  Thermometer 221 

164.  Velocity  of  a  Liquid  through  small  Orifice 224 

165.  Modification  due  to  Extraneous  Pressure 225 

166.  Spouting  of  Liquids  on  a  Horizontal  Plane 226 

167.  Coefficients  of  Efflux  and  Velocity 228 

168.  Efflux  through  Short  Tubes 230 

169.  Capillary  Phenomena 231 

170.  Elevation  and  Depression  between  Plates 232 

171.  Attraction  and  Repulsion  of  Floating  Bodies 233 

172.  Applications  of  the  Principles  of  Capillarity 234 

173.  Endosmose  and  Exosmose 235 

CHAPTER  Vni. 

MECHANICS   OF    GASES   AND   VAPORS. 

174.  Gases  and  Vapors 237 

175.  Atmospheric  Air 237 

176.  Atmospheric  Pressure 238 

177.  Mariotte's  Law 239 

178.  Gay  Lussac's  Law 242 

179.  Manometers 244 

180.  The  open  Manometer 244 

181.  The  closed  Manometer. 245 

182.  The  Siphon  Gauge 247 

183.  TheDiving-Bell 247 

184.  The  Barometer 248 


10  CONTENTS. 

Art,  PAGE 

185.  The  Siphon  Barometer 248 

186.  The  Cistern  Barometer 249 

187.  Uses  of  the  Barometer 250 

188.  Difference  of  Level 251 

189.  Steam 256 

190.  Work  of  Steam 258 

191.  Work  due  to  the  Expansion  of  a  Gas  or  Vapor 260 

CHAPTER  IX. 

HYDRAULIC    AND   PNEUMATIC   MACHINES. 

192.  Definitions 262 

193.  Water  Pumps 262 

194.  Sucldng  and  Lifting  Pump 262 

195.  Sucking  and  Forcing  Pump 267 

196.  Fire- Engine 270 

197.  The  Rotary  Pump 271 

198.  The  Hydrostatic  Press 272 

199.  Tlie  Siphon 275 

200.  The  Wurtemburg  Siplion 276 

201.  Tlie  Intermitting  Siphon 277 

203.  Intermitting  Springs 277 

203.  Siphon  of  Constant  Flow 277 

204.  The  Hydraulic  Ram 278 

205.  Archimedes'  Screw 280 

206.  The  Cliain  Pump 280 

207.  The  Air  Pump 281 

208.  Artificial  Fountains 284 

209.  Hero's  Ball 284 

210.  Hero's  Fountain 285 

211.  Wine-Taster  and  Dropping-Bottle 286 

212.  The  Atmospheric  Inkstand 287 

PRIME   MOVERS. 

213.  Definition  of  a  Prime  Mover 287 

214.  Water-wheels 288 

215.  Windmills 289 

216.  The  Steam-Eugine 290 

217.  Varieties  of  Steam-Engines 290 

218.  The  Boiler  and  its  Appendages 291 

219.  The  Engine  Proper 292 

220.  The  Locomotive 294 


MECHANICS 


CHAPTER  I. 

DEFINITIONS   AND   INTRODUCTORY    REMARKS. 
Definition  of  a  Body. 

1.  A  BODY  is  a  collection  of  material  particles.  A  body 
whose  dimensions  are  exceedingly  small  is  called  a  material 
point. 

Rest  and  Motion. 

2.  A  material  point  is  at  rest,  when  it  retains  a  fixed 
position  in  space ;  it  is  in  motion,  when  its  position  in  space 
is  continually  changing. 

Rest  and  motion,  with  respect  to  surrounding  objects, 
are  called  relative  rest  and  relative  motion,  to  distinguish 
them  from  absolute  rest  and  absolute  motion. 

Rectilinear  and  Curvilinear  Motion. 

3.  The  path  traced  out  by  a  moving  point  is  called  its 
trajectory.  When  this  is  a  straight  line,  the  motion  is 
rectilinear  ;  when  it  is  a  curve,  the  motion  is  curvilinear. 

When  the  motion  of  a  body  is  spoken  of  as  rectilinear  or 
curvilinear,  it  is  understood  that   some   particular  point 


12  MECHANICS. 

of  the  body  is  referred  to,  such  as  the  centre  of  gravity,  or 
the  centre  of  figujr^. , 

Motion  of  Translation  and  Rotation. 

-4/' A' *'66t^'y'  Imis  a  'maiicm  of  iranslation  when  all  its  points 

move  in  parallel  straight  lines;  it  has  a  motmi  of  rotation 

when  its  points  move  in  arcs  of  circles  having  their  centres 

in  the  same  line :  this  line  is  the  axis  of  rotation.    All  other 

varieties  of  motion  result  from  some  combination  of  these 

two. 

Uniform  and  Varied  Motion. 

5.  The  VELOCITY  of  a  point  is  its  rate  of  motion.  When 
it  moves  over  equal  spaces  in  equal  times,  the  velocity  is 
constant  and  the  motion  uniform;  when  it  moves  over 
unequal  spaces  in  equal  times,  the  velocity  is  variable  and 
the  motion  varied.  If  the  velocity  continually  increase, 
the  motion  is  accelerated ;  if  it  continually  decrease,  the 
motion  is  retarded. 

In  uniform  motion,  the  space  passed  over  in  one  second 
is  taken  as  the  measure  of  the  velocity.  In  varied  motion, 
the  velocity  at  any  instant  is  measured  by  the  space  that 
would  be  passed  over  in  a  second  were  the  velocity  to 
remain  the  same  as  at  that  instant. 

Definition  of  a  Force. 

6.  A  FORCE  is  anything  that  tends  to  change  the  state 
of  a  body  with  respect  to  rest  or  motion.  If  a  body  is  at 
rest,  anything  that  tends  to  put  it  in  motion  is  a  force ;  if 
a  body  is  in  motion,  anything  that  tends  to  change  either 
its  direction  or  its  rate  of  motion,  is  a  force. 

Classification  of  Forces, 

7.  Forces  may  be  divided  into  two  classes,  extraneous 
and  molecular:  extrancovs  forces  act  on  bodies  from  with- 


DEFIXITIOXS    A:N^U    INTRODUCTORiT    REMARKS.  13 

out;  molecular  forces  are  exerted  between  the  neighboring 
particles  of  bodies. 

Extraneous  Forces. 

8.  Extraneous  forces  are  of  two  kinds,  pressures  and 
moving  forces :  pressures  simply  tend  to  produce  motion ; 
moving  forces  actually  produce  motion.  Thus,  if  gravity 
act  on  a  fixed  body,  it  creates  pressure ;  if  on  a  free  body, 
it  produces  motion. 

Moving  forces  are  either  impulsive  or  incessant :  cm  im- 
pulsive force,  or  an  impulse,  is  one  that  acts  for  a  moment 
and  then  ceases;  an  incessant  force  is  one  that  acts  con- 
tinuously. We  may  regard  an  incessant  force  as  a  suc- 
cession of  impulses,  imparted  at  equal,  but  exceedingly 
small  intervals  of  time.  When  the  elementary  impulses 
are  equal,  the  force  is  constant;  when  they  are  unequal, 
the  force  is  variable.  Thus,  gravity,  at  any  place,  is  a 
constant  force ;  the  effort  of  expanding  steam  is  a  variable 
force. 

Molecular  Forces. 

9.  Molecular  forces  are  of  two  kinds,  attractive  and 
repellent :  attractive  forces  tend  to  bind  the  particles  of 
a  body  together;  repellent  forces  tend  to  thrust  them 
asunder.  Both  kinds  of  molecular  forces  are  continually 
exerted  between  the  molecules  of  bodies,  and  on  the  pre- 
dominance of  one  or  the  other  depends  the  physical  state 
of  a  body. 

Constitution  and  Classification  of  Bodies. 

10.  It  is  generally  believed  that  matter,  in  its  ultimate 
form,  consists  of  minute,  indivisible,  and  indestructible 
parts,  called  atoms.  These  are  grouped  in  various  ways, 
under  the  action  of  molecular  forces,  to  form  molecules,  or 


14  MECHANICS. 

particles;  and  these  again  are  united  to  form  larger 
bodies.  The  relations  that  exist  between  the  molecular 
forces,  in  different  cases,  form  a  basis  for  the  classifi- 
cation of  bodies:  they  are  divided  into  two  classes,  5o/i^6' 
and  fluids;  and  fluids  are  again  divided  into  liquids 
and  gases.  In  solids,  the  molecular  forces  of  attraction 
prevail  over  those  of  repulsion ;  in  liquids,  they  are 
nearly  balanced;  in  gases,  the  forces  of  repulsion  pre- 
vail over  those  of  attraction.  In  solids,  the  particles  ad- 
here so  as  to  require  some  force  to  separate  them;  in 
fluids,  the  particles  move  freely  amongst  each  other,  yield- 
ing to  the  slightest  force.  Solids  tend  to  preserve  both 
their  shape  and  volume:  liquids  tend  to  preserve  their 
volume,  but  take  the  shape  of  the  containing  vessel ;  gases 
have  no  tendency  to  retain  either  their  volume  or  their 
shape.  Many  bodies  are  capable  of  existing  in  different 
states,  according  to  temperature.  Thus,  ice,  loater,  and 
stea^n  are  the  same  body  in  different  states. 

Xissential  Properties  of  Bodies. 

11.  There  are  certain  properties  common  to  all  bodies, 
and  without  which  we  could  not  conceive  them  to  exist: 
these  are  extension,  impenetrahility,  and  inertia. 

ExTEKSioj^  is  that  property  by  virtue  of  which  a  body 
occupies  a  portion  of  space.  Every  body  has  length, 
breadth,  and  thickness. 

Impenetrability  is  that  property  by  virtue  of  which  no 
two  bodies  can  occupy  the  same  space  at  the  same  time. 
The  particles  of  one  body  may  be  thrust  aside  by  those  of 
another,  as  when  a  nail  is  driven  into  wood;  but  Avhere 
one  body  is,  no  other  body  can  be. 

Inertia  is  that  property  of  a  body  l)y  virtue  of  which  it 


DEFIXITIOXS    AKD    INTRODUCTORY    REMARKS.  15 

tends  to  continue  in  the  state  of  rest  or  motion  in  which  it 
may  be  placed,  until  acted  on  by  some  force. 

Matter  has  no  power  to  change  its  state  with  respect  to 
rest  or  motion ;  if  at  rest,  it  cannot  set  itself  in  motion ; 
or,  if  moving,  it  cannot  change  either  the  rate  or  the  direc- 
tion of  its  motion.  If  a  force  act  on  a  body  to  change  its 
state  of  rest  or  motion,  it  develops  a  resistance  that  acts  in 
a  contrary  direction.  This  resistance  is  called  the/orce  of 
inertia.  The  force  that  a  moving  body  possesses  and  is 
capable  of  giving  out,  when  its  motion  is  opposed,  is  called 
living  force. 

Laws  of  Motion. 

12.  The  laws  of  motion,  commonly  known  as  the  New- 
tonian Laws,  depend  on  the  principle  of  inertia.  They 
may  be  enunciated  as  follows : 

\st  Law.  If  a  body  be  at  rest,  it  will  remain  at  rest ; 
or  if  in  motion,  it  will  move  uniformly  in  a  straight  line, 
till  acted  on  by  some  force. 

2d  Lata.  If  a  body  be  acted  on  by  several  forces,  it  will 
obey  each  as  though  the  others  did  7iot  exist,  and  this  whether 
the  body  be  at  rest  or  in  motion. 

dd  Law.  If  a  force  act  to  change  the  state  of  a  body  with 
respect  to  rest  or  motion,  the  body  tvill  offer  a  resistance 
equal  and  directly  opposed  to  the  force. 

These  laws  are  deduced  from  universal  experience,  and 
are  accepted  as  axiomatic  in  treating  of  the  motion  of 
bodies. 

Secondary  Properties  of  Bodies. 

13.  Besides  the  properties  common  to  all  bodies,  there 
are  other  properties  possessed  in  a  greater  or  less  degree  by 
different  bodies,  that  may  be  called  secondary.  Of  these, 
the  most  important,  from  a  mechanical  point  of  view,  are 


16  MECHANICS. 

porosity,  compressibility,  dilat ability,  and  elasticity,  all  of 
which  arise  from  peculiarity  of  atomic  constitution. 

POKOSITY  is  that  property  by  virtue  of  which  the  par- 
ticles of  a  body  are  more  or  less  separated.  The  interme- 
diate spaces  are  called  pores.  When  the  pores  are  small, 
the  body  is  dense  ;  when  they  are  large,  it  is  rare.  Gold  is 
a  dense  body,  hydrogen  a  rare  one.  It  is  to  be  observed  that 
the  interatomic  spaces,  which  are  properly  called  pores,  are 
regularly  distributed  throughout  the  body,  and  should  not 
be  confounded  with  those  irregular  spaces  that  may  be 
called  cavities  or  cells,  examples  of  which  may  be  seen  when 
a  loaf  of  bread  is  cut  across. 

Compressibility  is  that  property  by  virtue  of  which  the 
particles  of  a  body  may  be  made  to  approach  each  other, 
so  as  to  occupy  less  space. 

DiLATABiLiTY  is  that  property  by  virtue  of  which  the 
particles  of  a  body  may  be  separated  to  a  greater  distance, 
so  as  to  occupy  more  space. 

Elasticity  is  that  property  by  virtue  of  which  a  body 
tends  to  resume  its  original  form,  or  volume,  after  com- 
pression or  extension.  The  effort  that  a  body  exerts  to 
return  to  its  original  form  or  volume  after  distortion,  is 
called  the  force  of  restitution;  and  when  this  is  very  great 
in  comparison  with  the  force  of  distortion,  the  body  is 
highly  elastic.  Ivory  is  an  example  of  a  highly  elastic  body ; 
clay  is  very  inelastic.  Within  certain  limits  most  bodies 
may  be  considered  as  elastic, — that  is,  if  they  be  slightly 
distorted,  they  will  completely  recover  their  original  shape, 
or  volume,  on  the  removal  of  the  force  of  distortion. 

Force  of  Gravity  and  Weight, 

14.  Observation  shows  that  the  earth  exercises  an  at- 
tractive force  on  bodies,  tending  to  draw  them  toward  its 


DEFIXITIOXS   XND    INTRODUCTOKY    REMAKKS.  17 

centre.  This  force  is  called  the  force  of  gi^avity.  It  acts 
on  every  particle,  and  if  the  body  be  supported,  it  produces 
a  pressure  proportional  to  the  quantity  of  matter  in  it; 
this  pressure  is  called  the  lueight  of  the  body. 

Newton  showed  that  terrestrial  gravity  is  only  a  particu- 
lar manifestation  of  a  general  law,  which  certainly  prevails 
throughout  the  solar  system,  and  probably  throughout  the 
physical  universe.  This  law,  sometimes  called  the  New- 
tonian law  of  imiversal  gravitation,  may  be  enuuciated  as 
follows : 

Every  iMrtide  of  matter  attracts  every  other  particle, 
with  a  force  that  varies  directly  as  the  mass  of  the  attract- 
ing particle,  and  inversely  as  the  square  of  the  distance 
between  the  i)articles. 

It  has  also  been  shown  that  the  attraction  of  the  earth 
on  bodies  exterior  to  it,  is  very  nearly  the  same  as  though 
all  its  matter  were  concentrated  at  its  centre.  Because  the 
form  of  the  earth  is  that  of  an  oblate  spheroid,  having  its 
axis  coincident  with  that  of  revolution,  the  force  of  gravity 
increases  slighty  in  passing  from  the  equator  toward  the 
pole.  The  weight  of  a  body  must  therefore  increase  at  the 
same  rate.  That  this  increase  of  weight  may  be  rendered 
apparent,  the  weighing  must  be  performed  by  a  spring 
balance,  or  some  equivalent  method,  for,  were  the  ordinary 
balance  used,  the  increased  weight  of  the  body  would  be 
accompanied  by  a  like  increase  in  the  weight  of  the  coun- 
terpoise. 

Mass  and  Density. 

15.  The  MASS  of  a  body  is  the  quantity  of  matter  it  con- 
tains. We  have  seen  that  the  weight  of  a  body  increases 
at  the  same  rate  as  the  force  of  gravity ;  hence  the  quo- 
tient obtained  by  dividing  the  weight  at  any  place  by  the 


18  MECHANICS. 

force  of  gravity  at  that  place  is  constant.  This  quotient  is 
always  proportional  to  the  quantity  of  matter  in  the  body, 
and  for  this  reason  is  taken  as  the  measure  of  its  mass. 
Denoting  the  mass  of  a  body  by  M,  its  weight  by  W,  and 
the  force  of  gravity  by  g,  we  have, 

W 
M  =  —',  whence,  W=  Mg (1) 

The  DENSITY  of  a  body  is  the  degree  of  compactness  of 
its  particles.  It  is  proportional  to  the  quantity  of  matter 
in  a  given  volume.  We  may  take,  as  the  measure  of  a 
body's  density,  the  quotient  of  its  mass  by  its  volume; 
or,  denoting  the  density  by  D,  the  volume  by  F,  and  the 
mass  by  M,  we  have, 

Combining  this  with  equation  (1),  we  find, 

D  =  ^',  whence,  W  =  DVg (2) 

Formulas  (1)  and  (2)  are  of  frequent  use  in  Mechanics. 

The  quantity  of  matter  that  weighs  one  pound  is  taken 
as  the  unit  of  mass.  The  density  of  distilled  water  at  39"^ 
Fah.  is  taken  as  the  U7iit  of  density. 

Momentum  or  Quantity  of  Motion. 

16.  The  MOMENTUM,  or  the  quantity  of  motion  of  a 
body,  is  the  product  of  its  mass  by  its  velocity.  If  a  force 
act  to  impart  motion,  it  is  obvious  that  the  force  must  in 
the  first  place  be  proportional  to  the  mass  moved ;  and  in 
the  second  place,  to  the  velocity  it  can  impart.  It  is  in 
accordance  with  this  principle  that  momentum,  or  quantity 
of  motion,  is  used  as  a  measure  of  force. 


DEFINITIONS   AND   INTRODUCTORY   REMARKS.  19 

Measure  of  Forces. 

17.  A  force  is  measured  by  comparing  it  with  some  other 
force  of  tlie  same  kind  taken  as  a  unit.  There  are  two 
kinds  of  forces — pressures  and  moving  forces ;  and  conse- 
quently two  kinds  of  units. 

The  unit  of  pressure  is  one  pound ;  when  we  speak  of  a 
pressure  of  n  pounds,  we  mean  a  force  that  would,  if  di- 
rected vertically  upward,  just  sustain  a  weight  of  n  pounds. 

The  unit  of  an  impulsive  force  is  an  impulse  capable  of 
imparting  a  unit  of  velocity  to  a  unit  of  mass ;  that  is,  an 
impulse  capable  of  generating  a  unit  of  momentum. 

An  impulsive  force  is  measured  by  the  quantity  of  mo- 
tion it  can  generate.  If  an  impulse  /  impart  a  velocity  v 
to  a  mass  7n,  we  have, 

f=mv (3) 

Impulses  acting  on  the  same  or  on  equal  masses,  are  pro- 
portional to  the  velocities  they  impart. 

The  unit  of  a  constant  force  is  a  constant  force  capable 
of  generating  a  unit  of  momentum  in  a  unit  of  time. 

A  constant  force  is  measured  by  the  quantity  of  motion 
it  can  generate  in  a  unit  of  time.  If  a  constant  force  / 
generate  a  quantity  of  motion  equal  to  7nv  in  a  unit  of  time, 
we  have, 

f=mv (4) 

Constant  forces  acting  on  equal  masses  are  proportional 
to  the  velocities  they  generate  in  the  same  time. 

We  have  seen  that  an  incessant  force  may  be  regarded 
as  a  succession  of  impulses,  imparted  at  equal  intervals  of 
time  (Art.  8) ;  hence,  constant  forces  are  proportional  to 
their  elementary  impulses. 

Variable  forces  have  different  values  at  different  times. 


20  MECHANICS. 

The  measure  of  such  a  force,  at  any  instant,  is  the  quantity 
of  motion  it  could  generate  in  a  unit  of  time,  if  its  intensity 
were  to  remain  unchanged  for  that  time.  The  values  of 
variable  forces  at  different  times  are  proportional  to  their 
elementary  impulses  at  those  times. 

Acceleration  due  to  a  Force. 

18.  The  velocity  that  a  constant  force  can  generate  in  a 
body  in  a  unit  of  time,  is  called  the  acceleration  due  to  the 
force.  If  we  find  the  value  of  v,  in  equation  (4)  of  the  last 
article,  we  have, 


7)1 


(5) 


That  is,  the  acceleration  is  equal  to  the  moving  force,  divided 
hy  the  mass  moved. 

If  the  acceleration  is  known,  the  moving  force  may  be 
found  by  multiplying  the  acceleration  by  the  mass.  In 
some  cases  the  force  acts  independently  on  each  ]3article; 
the  acceleration  is  then  independent  of  the  mass.  The 
force  of  gravity  is  an  example  in  which  the  acceleration  is 
independent  of  the  mass. 

Representation  of  Forces. 

19.  Forces  may  be  represented  geometrically  by  straight 

lines,  proportional  to  the  forces.     A  force  is  given  when  we 

know  its  intensity,  its  point  of  application,  and  the  direction 

in  tuhicJi  it  acts.     When  a  force  is  represented  by  a  line, 

the  length  of  the  line  represents  its  in-      ,^_____ 

tensity ;    one   extremity  represents    the     0  P 

point  of  application ;  and  an  arrow-head 
at  the  other  extremity  shows  the  direction  of  the  force. 
Thus,  in  Figure  1,  OP  is  the  intensity  of  the  force ;  0,  its 


DEFINITIONS    AXJ)    INTRODUCTOKT   REMARKS.  21 

point  of  application;  and  OP,  the  direction  in  which  it 
acts.  If  a  force  be  applied  to  a  solid  body,  the  point  of 
application  may  be  taken  anywhere  on  its  line  of  direc- 
tion; and  it  is  often  found  convenient  to  transfer  it  from 
one  point  of  this  line  to  another.  The  line  OP  prolonged 
indefinitely  is  called  the  line  of  action  of  the  force  OP. 

A  force  may  be  represented,  analytically,  by  a  letter;  thus 
the  force  OP  may  be  called  the  force  P.  In  this  case  we 
assume  the  usual  algebraic  rule  for  estimating  quantities; 
that  is,  if  a  quantity  in  one  sense  is  i^ositive,  a  quantity 
in  an  opposite  sense  must  be  negative. 

Equilibrium. 

20.  Forces  are  in  equilibrium  when  they  balance  each 
other.  If  a  system  of  forces  in  equilibrium  be  applied  to  a 
body,  they  will  not  change  its  state  with  respect  to  rest 
or  motion  :  if  the  body  be  at  rest,  it  will  remain  so ;  if  in 
motion,  its  motion  will  remain  unchanged,  so  far  as  these 
forces  are  concerned. 

When  forces  balance  each  other  through  the  medium  of 
a  body  at  rest,  they  are  said  to  be  in  statical  equilihrium  ; 
when  they  balance  each  other  through  tlie  medium  of  a 
moving  body,  they  are  in  dyyiarnical  equilihrium. 

If  a  body  be  at  rest,  or  if  in  uniform  motion,  we  con- 
clude that  the  forces  acting  on  it  are  in  equilibrium. 

Definition  of  Mechanics. 

21.  Mechanics  is  the  science  that  treats  of  the  action 
of  forces  on  bodies. 

It  treats  of  the  laws  of  equilibrium  and  motion,  and  is 
sometimes  divided  into  two  branches,  called  Statics  and 
Dynamics.     Statics  treats  of  pressures ;  Dynamics,  of  mov- 


22  MECHANICS. 

ing  forces:  when  the  bodies  acted  on  are  liquids,  these 
branches  are  called  hydrostatics  and  hydrodynamics ; 
when  the  bodies  acted  on  are  gases,  they  are  called  aero- 
statics and  aerodynamics. 

A  better  division  of  the  subject  is  into  mechanics  of  solids 
and  mechanics  of  fluids. 


CHAPTER  11. 

COMPOSITION^,  RESOLUTIOIS^,  AKD  EQUILIBRIUM  OF  FORCES. 

Definition. 

22.  Composition  of  forces,  is  the  operation  of  finding  a 
single  force  whose  effect  is  the  same  as  that  of  two  or  more 
given  forces.  The  required  force  is  called  the  resultant  of 
the  given  forces. 

Resolution  of  forces,  is  the  operation  of  finding  two  or 
more  forces  whose  combined  effect  is  equivalent  to  that  of 
a  given  force.  The  required  forces  are  called  components 
of  the  given  force. 

Composition  of  Forces  whose  directions  coincide. 

23.  From  the  rules  laid  down  for  measuring  forces,  it 
follows,  that  the  resultant  of  two  forces  applied  at  a  point, 
and  acting  in  the  same  direction,  is  equal  to  the  sum  of  the 
forces.  If  two  forces  act  in  opposite  directions,  their  result- 
ant is  equal  to  their  difference,  and  it  acts  in  the  direction 
of  the  greater. 

If  any  number  of  forces  be  applied  at  a  point,  some 
in  one  direction,  and  others  in  a  contrary  direction,  their 
resultant  is  equal  to  the  sum  of  those  that  act  in  one 
direction,  diminished  by  the  sum  of  those  that  act  in  the 
opposite  direction ;  or,  if  we  regard  the  rule  for  signs,  the 
resultant  is  equal  to  the  algebraic  sum  of  the  components  ; 
the  sign  of  this  sum  indicates  the  direction  in  which  the 
resultant  acts.    Thus,  if  the  forces  P,  P',  &c.,  act  on  a  point, 


24  MECHANICS. 

and  in  a  direction  that  we  may  assume  as  positive,  whilst 
the  forces  P",  JP'",  &c.,  act  on  the  same  point  and  in  the 
opposite  direction,  then  will  their  resultant,  denoted  by  E, 
be  given  by  the  equation, 

72  =  (P  +  P'  +  &c,)  —  (P"  +  P'"  +  &c.) 

If  the  first  term  of  the  second  member  is  numerically 
greater  than  the  second,  B  is  positive,  and  the  resultant 
acts  in  the  direction  that  we  assumed  as  positive ;  if  the 
first  term  is  numerically  le^s  than  the  second,  R  is  negative, 
and  the  resultant  acts  in  the  opposite  direction ;  if  the  two 
terms  are  equal,  the  resultant  is  0,  and  the  forces  are  in 
equilibrium. 

All  the  forces  of  a  system  that  act  in  the  general  direc- 
tion of  any  straight  line,  are  called  homologous,  and  their 
algebraic  sum  may  be  expressed  by  writing  the  expression 
for  single  force,  prefixing  the  symbol  2,  which  indicates 
the  algebraic  siwi  of  homologotis  quantities.  We  might,  for 
example,  write  the  preceding  equation  under  the  form, 

^  =  2  (^) (6) 

This  equation  expresses  the  fact,  that  the  resultant  of  a 
system  of  homologous  forces,  is  equal  to  their  algebraic 
sum. 

Composition  of  concurrent  Forces. . 

24.  Concurrent  forces  are  those  whose  lines  of  direction 
intersect  in  a  common  point.  The  simplest  case  is  that  of 
two  forces  applied  at  a  common  point,  but  not  in  the  same 
direction.  After  this,  in  order  of  simplicity,  we  have  the 
case  of  several  forces  applied  at  a  common  point  and  lying 
in  the  same  plane,  and  then  the  case  of  several  forces  ap- 
plied st  a  common  point  and  not  in  a  single  plane. 


COMPOSITION,  ETC.,  OF   FORCES.  25 

Parallelogram  of  Forces. 

25.  Let  0  be  a  material  point,  and  suppose  it  acted  on 
by  two  simultaneous  impulses,  P  and  Q,  represented  in 

direction   and   intensity   by  OP  and        q  _  H 

OQ;  complete  the  parallelogram  P$,         /  ^^i 

and  draw  its  diagonal  OR,  I   ^^ 

If  0  be  taken  as  the  unit  of  mass,    i^^^ ^ 

OP  and  OQ  will  represent  the  veloci-  ^ 
ties  due  to  Pand  Q,  (Art.  17),  and  in- 
asmuch as  the  point  obeys  each  force,  as  though  the  other 
did  not  exist,  (Art.  12),  it  will  be  found  at  the  end  of  one 
second  somewhere  on  PR,  by  virtue  of  the  force  P,  and 
somewhere  on  QR,  by  virtue  of  the  force  Q  ;  it  will  there- 
fore be  at  R,  and  because  it  moves  uniformly  in  the  direc- 
tion of  each  force,  it  must  move  uniformly  in  the  direction 
OR.  Had  O  been  acted  on  by  an  impulse  represented 
by  OR,  it  would  in  like  manner  have  moved  uniformly 
from  0  to  R  in  one  second.  Hence  the  impulse  OR  is 
equivalent  in  effect  to  the  two  impulses  OP  and  OQ  ; 
that  is. 

If  kvo  impulsive  forces  be  represented  hy  adjacent  sides 
of  a  parallelogram,  their  resultant  will  he  represented  hy 
that  diagonal  of  the  parallelograin  ivhich  passes  through 
their  common  point. 

Because  constant  forces  are  proportional  to  their  elemerd- 
ary  impulses,  (Art.  17),  the  above  principle  holds  true  for 
them ;  and  because  variable  forces  are  measured  by  sup- 
posing them  to  become  constant  for  a  unit  of  time,  (Art. 
17),  the  principle  must  hold  true  for  them:  it  is  therefore 
true  for  all  kinds  of  moving  forces.  It  is  also  true  for 
forces  of  pressure,  for  if  we  apply  a  force  equal  and  directly 
opposed  to  the  resultant  of  two  moving  forces,  it  will  hold 


2:6  MECHANICS. 

them  ill  equilibrium,  converting  them  into  forces  of  pres- 
sure, but  it  will  in  no  manner  change  the  relation  between 
them  and  their  resultant.  Hence,  the  principle  is  univer- 
sal ;  it  may  be  enunciated  as  follows : 

If  Uoo  forces  be  represented  in  direction  and  inteyi.^itij 
by  adjacent  sides  of  a parallelogravi,  their  resultant  will 
be  represented  by  that  diagonal  of  the  parallelogram  which 
passes  through  their  common  point. 

This  principle  is  called  the  parallelogram  of  forces. 

Geometrical  Applications  of  the  Parallelogram  of  Forces. 

26.  1°.  Given    two   forces;  to   find  ^ 

their  resultant.  y ""^^ 

Let  OP  and  OQho;  the  given  forces.        /      ^^    / 
Complete  the  parallelogram  QP  and      1^^ 
draw  its  diagonal   OR ;  this  will  be    0  ^ 

the  resultant  required.  ^^^'  ^' 

2°.  Given,  a  force  and  one  of  its  components ;  to  find 
the  other. 

Let  OR  be  a  force  and  OP  one  of  its  components. 
Draw  PR  and  complete  the  parallelogram  PQ  ;  OQ  will 
be  the  other  component. 

3°.  Given,  a  force  and  the  directions  of  its  components ; 
to  find  the  components. 

Let  OR  be  a  force  and  OP,  OQ,  the 
directions  of  its  components ;  through 

R  draw   RQ  and   RP  parallel  to  PO  ^<g^^- ^ 

and  QO  ;  then  will  OP  and  OQhe  the  Fig.  4. 

required  components. 

4°.  Given,  a  force  and  the  intensities  of  its  com.po- 
nents ;  to  find  the  directions  of  the  components. 

Let  OR  be  a  force,  and  let  the  intensities  of  its  com- 
ponents be  represented  by  lines  equal  to  0/*and  OQ  ;  with 


COMPOSITION,  ETC.,  OF    FORCES. 


2? 


Fig.  5. 


0  as  a    centre  and    OP    as   radius, 

describe  an  arc,    then  with  i?  as   a 

centre  and  OQ  as  a  radius,  describe 

a  second  arc,  cutting  the  first  at  P  ; 

draw    OP,   and   RP,  and   complete 

the  parallelogram  PQ ;   OP  and  OQ  will  be  the  required 

components. 

Polygon   of  Forces. 

27.  Let  OQ,  OP,  OS,  and  OT,  be 

a  system  of  forces  applied  at  a  point, 
0,  and  lying  in  a  single  plane.  To 
construct  their  resultant ;  on  OQ 
and  OP  construct  the  parallelogram 
PQ,  and  draw  its  diagonal  OR',  this 
will  be  the  resultant  of  OP  and  OQ. 
In  like  manner  construct  a  parallel- 
ogram on  OjR' and  OS;  its  diag-  ^^s-^- 
onal  OR",  will  be  the  resultant  of  OP,  OQ,  and  OS, 
On  OR"  and  OT  construct  a  parallelogram,  and  draw 
its  diagonal  OR  ;  then  will  OR  be  the  resultant  of  all 
the  given  forces.  This  method  of  construction  may  be 
extended  to  any  number  of  forces  whatever. 

If  we  examine  the  diagram,  we  see  that  QR'  is  parallel 
and  equal  to  OP,  R' R"  is  parallel  and  equal  to  OS,  R" R 
is  parallel  and  equal  to  OT,  and  that  OR  is  drawn  from 
the  point  of  application,  0,  to  the  extremity  of  R"R. 
Hence,  we  have  the  following  rule  for  constructing  the 
resultant  of  several  concurrent  forces : 

Through  their  common  point  draw  a  line  parallel  and 
equal  to  the  first  force  ;  through  the  extremity  of  this  draw 
a  line  parallel  and  equal  to  the  second  force ;  and  so  on, 
throughout  the  system  ;  finally,  draw  a  line  from  the  start- 


28  MECHANICS. 

ing  point  to  the  extremity  of  the  last  line  drawn,  and  this 
toill  be  the  res2ilta7it  required. 

This  application  of  the  parallelogram  of  forces,  is  called 
the  polygon  of  forces. 

The  principle  holds  true,  even  when  the  forces  are  not 
in  one  plane.  In  this  case,  the  lines  OQ,  QR',  R' R",  &c., 
form  a  tivisted  polygon,  that  is,  a  polygon  whose  sides  are 
not  in  one  plane. 

When  the  point  R,  in  the  construction,  falls  at  0,  OR 
reduces  to  0,  and  the  forces  are  in  equilibrium. 

Parallelopipedon  of  Forces. 

28.  Let  OP,  OQ,  and  OS,  be  three  concurrent  forces  not 
in  the  same  plane.  On  these,  as  edges,  construct  the 
parallelopipedon    OR,  and   draw    OR, 

OM,  and  SR.     From  the  principle  of  J.;- -,-, 

Art.  25,   OM  is  the  resultant  of  OP     ^i^___^^^''  \ 
and  OQ  J  and  OR  is  the  resultant  of     |     tA^--^^"^  r  y}.^ 
OM  and  OS;  hence,  Oi?  is  the  result-     |^/^       "l::::^!'''' 

ant  of  OP,   OQ,  and  OS;  that  is,  if     Q" "    ^ 

three  forces  be  represented  by  the  con-  ^' 

current  edges  of  a  parallelopipedon,  their  resultant  will  be 
represented  by  the  diagonal  of  the  parallelopipedon  that 
passes  through  their  common  point.  This  principle  is 
called  the  parallelopipedon  of  forces.  It  is  easily  shown 
that  it  is  a  particular  case.of  the  polygon  of  forces  ;  for,  OP 
is  parallel  and  equal  to  the  first,  PM  to  the  second,  MR 
to  the  third  force,  and  OR  is  drawn  from  the  origin,  0,  to 
the  extremity  of  MR. 

Components  of  a  force  in  the  direction  of  Rectangular  Axes. 

29.  First.  To  find  analytical  expressions  for  the  com- 
ponents of  a  force  in  the  direction  of  two  axes. 


COMPOSITION",  ETC.,  OF   FORCES. 


29 


Fijj.  S. 


Let  AE  he  n  force  in  the  plane  of 
the  rectangular  axes  OX  and  OY.  On 
it  as  a  diagonal  construct  a  parallelo- 
gram ML,  whose  sides  are  parallel  to 
OX  and  OY.  Denote  AR  hy  E,  AL 
by  X,  AM,  equal  to  LE,  by  Y,  and 
the  angle  LAE,  equal  to  the  angle 
the  force  makes  with  OX,  by  a.  From  the  figure,  we 
have, 

X  —  E  cos  a,  and  F  =  7?  sin  a (7) 

In  these  expressions  the  angle  a.  is  estimated  from  the 
positive  direction  of  the  axis  of  X,  around  to  the  force,  in 
accordance  with  the  rule  laid  down  in  Trigonometry.  The 
component  X  will  have  the  same  sign  as  cos  a,  and  the 
component  Y  the  same  sign  as  sin  a. 

Secondly.  To  find  the  components  of  a  force  in  the  direc- 
tion of  three  rectangular  axes. 

Let  OE,  denoted  by  E,  be  the  given  force,  and  OX,  OY, 
and  OZ,  the  given  axes.  On  OE,  as  a  diagonal,  construct 
a  parallelopipedon  whose  edges 
are  parallel  to  the  axes.  Then 
will  OL,  OM,  and  OiV^be  the  re- 
quired components.  Denote  these 
by  X,  Y,  and  Z,  and  the  angles 
they  make  with  OE  by  a,  (3,  and 
7.  Join  E  with  L,  M,  and  X,  by 
straight  lines.  From  the  right- 
angled  triangles  thus  formed,  we 
have, 


Fig.  9. 


X  =  E  cos  a,  Y  =  E  cos  (3,  and  Z  =  E  cos  y  .  .  .  (8) 
The  angles  a,  [3,  and  7  are  estimated  from  the  positive 


30  MECHANICS. 

directions  of  tlie  corresponding  axes,  as  in  Trigonometry, 
and  eacli  component  has  the  same  sign  as  the  correspond- 
ing cosine. 

If  a  force  be  resolved  in  the  direction  of  rectangular 
axes,  eacli  component  will  represent  the  total  effect  of  the 
given  force  in  that  direction.  For  thij  reason  such  com- 
ponents are  called  effective  components.  It  is  plain,  that 
the  component  in  the  direction  of  each  axis,  is  the  same  as 
the proJectio7i  of  the  force  on  that  axis,  the  projection  being 
made  by  lines  through  the  extremities  of  the  force,  and 
perpendicular  to  the  axis.  Hence,  we  may  find  the  effective 
component  of  a  force  in  the  direction  of  a  given  line, 
geometrically,  by  projecting  the  force  on  the  line,  or  a?i- 
alytically,  by  multiplying  the  force  into  the  cosine  of  its 
inclination  to  the  line. 


Analytical  Composition  of  Rectangular  Forces 

30.  First.    When   there   are   but  two 
forces. 

Let    AL    and    AM    be    rectangular 
forces,  denoted   by  X  and   Y,   and   let 
AR,  denoted  by  7?,  be  their  resultant.  - 
Denote   the   angle   RAL  by  a.     Then, 
because   LR  =  Y,   we    have,   from   the  Fig.  lo 

triangle  ALR, 


.CL    1 


X         .    .  Y 


R  =v^X^  +  i  ';  cos  ct  =  —  ;  and  sin  a  =  ~ (9) 

The  first  of  these  gives  the  intensity,  the  second  and  third 
the  direction  of  the  resultant. 

Secondly.  When  there  are  three  forces  not  in  one  plane. 
Let  OL,  OM,  and  ON,  be  rectangular  forces  denoted  by 
A",  Y,  and  Z,  and  let  OR,  denoted  1)V  /?,  be  their  result- 


COMPOSITION,  ETC.,  OF   FORCES. 


31 


ant.  Denote  the  angles  which 
R  makes  with  OL,  OM,  and  OJV 
by  a,  13,  and  7.  Then,  from  the 
figure,  we  have, 

E  =  ^X^  +  Y 

X 

cos  a  =  —  ;  COS 
K 

COS  7  = 


Fig.  n. 


The  first  gives  the  intensity  of  the  resultant,  the  others 
its  direction. 

Examples. 

1.  Two  pressures  of  9  and  12  pounds  act  on  a  point,  and  at  right 
angles  to  each  other.    Required,  the  resultant  pressure. 

SOLUTION. 

We  have, 

X=9,  and  F=:12; 
9 
Also,  cos  a  =  —  =  .6 ; 
15 

That  is,  the  resultant  pressure  is  15  lbs.,  and  it  makes  an  angle  of 
53°  7  47"  with  the  first  force. 

2.  Two  rectangular  forces  are  to  each  other  as  3  to  4,  and  their 
resuitant  is  20  lbs.     What  are  the  intensities  of  the  components  ? 

SOLUTION. 

We  have,  3F=  4  X,  or  Y-  \  X,  and  i?  =  20; 


.-.  i2=v/81  +  144  =  15. 

.-.  a  =  53°  7'  47". 


Hence,  X  =  12,  and  Y=  16. 

3.  A  boat  fastened  by  a  rope  to  a  point  on  shore,  is  urged  by  the 
wind  perpendicular  to  the  current,  with  a  force  of  18  pounds,  and 
down  the  current  by  a  force  of  22  pounds.  What  is  the  tension  on 
the  rope,  and  what  angle  does  it  make  with  the  cuiTent? 

SOLUTION. 

We  have, 

A'  =  22,  and  F  =  18 ;  .:  E  =Vm  =  28.425 ; 


Al  22 

Also,      eoS«  =  — -: 


a  =  39°  IT  20". 


32  MECHAXrCS. 

Hence,  the  tension  is  28.425  lbs.,  and  the  angle  39°  17'  20". 

4.  Required  the  intensity  and  direction  of  the  resultant  of  three 
forces  at  right  angles  to  each  other,  having  tlie  intensities  4,  5,  and  6 
pounds,  respectively. 

SOLUTION. 

We  have, 

X  =  4,  F  =  5,  andZ=6.  .-.  22  =  >/77  =  8.775. 

Also,  cos  a  =^_,  cos  /?  =  ^-A_,  and  cos  y  =  -^.; 

whence,  a  ~  02°  52'  51",  f5  =  55°  15'  50",  and  y  =  46°  51'  43". 

Hence  the  resultant  pressure  is  8.775  lbs.,  and  it  makes,  with  the 
components  taken  in  order,  angles  equal  to  62°  52'  51",  55°  15'  50", 
and  46°  51'  43  ". 

5.  Three  forces  at  right  angles  are  to  each  other  as  2,  3,  and  4,  and 
their  resultant  is  60  lbs.     What  are  the  intensities  of  the  forces  ? 

Am.  22.284  lbs.,  33.426  lbs.,  and  44.568  lbs. 

Application  to  Groups  of  Concurrent  Forces. 

31.  The  principles  explained  in  the  preceding  arti- 
cles, enable  us  to  find  the  resultant  of  any  number  of 
concurrent  forces.  Let  P,  P',  P",  &c.,  be  a  group  of  con- 
current forces.  Call  the  angles  they  make  with  the  axis 
of  JT,  a,  a',  a",  &c. ;  the  angles  they  make  with  the  axis  of 
Y,  (3,  i3',  ^",  &c. ;  and  the  angles  they  make  with  the  axis 
of  Zy  y,  y,  y",  &c.  Resolve  each  force  into  rectangular 
components  parallel  to  the  axes,  and  denote  the  resultants 
of  the  groups  parallel  to  the  axes  by  X,  Y,  and  Z.  We 
have,  (Art.  23), 

X=I{P  cos  a),  Y  =  I{P  cos  iS),  Z=  1'{P  cos  7). 

If  we  denote  the  resultant  by  R,  and  the  angles  it 
makes  with  the  axes  by  a,  h,  and  c,  w^e  have,  as  in  Arti- 
cle 30, 


X  Y  Z 

cos  a  =  1-,  cos  b  =  —,  and  cos  c  =  —. 
R  R  R 


COMPOSITION",  ETC.,  OF   FORCES.  33 

These  formulas  determine  the  intensity  and  direction  of 
the  resultant. 

When  the  given  forces  lie  in  the  plane  Xl",  Z  reduces 
to  0,  cos  (i  becomes  sin  a,  cos  h  becomes  sin  a,  and  the  for- 
mulas reduce  to, 

X  =  I  {F  cos  a),  and  Y  ~  ^  {F  sin  a). 

X  T 

R  —  v/X^  4-  I^,  and  cos  a  =  jy,  and  sin  a  =  -jr. 

Ji  K 

Examples. 

1.  Three  concurrent  forces,  whose  intensities  are  50,  40,  and  70,  lie 
in  the  same  plane,  and  make  with  an  axis,  angles  equal  to  15°,  30°, 
and  45°.     Required  the  resultant. 


SOLUTION. 

We  have. 

X  = 

5p 

COS  15° 

+  40 

COS  30°  +  70 

cos  45°  : 

=  133.435, 

and 

T= 

:50 

sin  15° 
R 

+  40 

sin  30°  +  70 

sin  45° : 
=  156 ; 

=  82.44 ; 

whence 

(98  +  17539  = 

132.4R5 

and. 

COS  a 

1 

.n     ;     ••  «  = 

=  31°  54' 

12". 

156 

The  resultant  is  156,  and  the  angle  it  makes  with  the  axis  is  31° 
54'  12". 

2.  Three  forces  4,  5,  and  0,  lie  in  the  same  plane,  and  make  equal 
angles  with  each  other.  Required  the  intensity  of  their  resultant 
and  the  angle  it  makes  with  the  least  force. 

SOLUTION. 

Take  the  least  force  as  the  axis  of  X     Then  the  angle  between  it 

and  the  second  force  is  120°,  and  that  between  it  and  the  third  force 

is  240°.     We  have, 

X=  4  +  5  cos  120°  +  6  cos  240°  =  -  1.5 ; 

r=  5  sin  120°  +  6  sin  240°  =  -  .866 ; 

/o  1-5        .  .866  ^,^„ 

.-.  R  =  V3,  cos  a  =  -  p^ ,  sm  a  =  -  ^^ ;     .-.   a  =  210'. 

3.  Two  forces,  one  of  5  lbs.  and  the  other  of  7  lbs.,  are  applied  at 
the  same  point,  and  make  with  each  other  an  angle  of  120°.  What 
is  the  intensity  of  their  resultant.     Arts.  6.24  lbs. 


34 


MECHANICS. 


Formula  for  the  Resultant  of  two  Forces. 

32.  Let  P  and  P\  be  two  forces  in  the  same  plane,  and 
let  the  axis  of  Xbe  taken  to  coincide  with  P;  a  will  then 
be  0,  and  we  shall  have  sin  a  =  0,  and 
cos  a  =  1.  The  value  of  X  (Art.  31) 
will  be  P  -\-  P'  cos  a',  and  the  value 
of  J"  will  be  P'  sin  a'.  Squaring 
these  values,  substituting  in  Equa- 
tion (9),  and  reducing  by  the  relation 
sin"  a!  +  cos'  a'  ==  1,  we  have. 


Pig.  12. 


R=Vp'  +  P'^  +  2PP'  cos  a' (12) 

The  angle  a'  is  the  angle  between  the  given  forces. 
Hence, 

Tlie  resultant  of  two  concurrent  forces  is^  equal  to  the 
square  root  of  the  sum  of  the  squares  of  the  forces,  plus 
twice  the  product  of  the  forces  into  the  cosine  of  their  in- 
cluded angle. 

If  a'  is  greater  than  90°,  and  less  than  270°,  its  cosine  is 
negative,  and  we  have. 


R  =Vp^  +  p'^  -  2PP'  cos  a'. 
If  a!  =  0,  its  cosine  is  1,  and  we  have, 

R  =  P  +  P', 
If  a  =  90°,  its  cosine  is  0,  and  we  have, 


R  r=  N/p«  +  P'\ 
If  a'  =  180°,  its  cosine  is  —  1,  and  we  have, 
R  =  P-  P'. 

Examples. 

1.  Two  forces,  P  and  Q,  are  equal  to  24  and  30,  and  the  angle  "be- 
tween them  is  105°.     What  is  the  intensity  of  their  resultant? 


E  =  %/2i^  +  80'  +  2  X  24  X  30  cob  lOS**  =  83.21. 


COMPOSITION,  ETC.,  OF   FORCES.  35 

2.  Two  forces,  P  and  Q,  whose  intensities  are  5  and  13,  have  a 
resultant  whose  intensity  is  13.     Required  the  angle  between  them. 


13  =\/25  +  144  +  2  X  5  X  12  cos  a. 

.-.  cos  a  =  0,  or  n:  =  90°,     An». 

3.  A  boat  is  impelled  by  the  current  at  the  rate  of  4  miles  per  hour, 
and  by  the  wind  at  the  rate  of  7  miles  per  hour.  What  will  be  her 
rate  per  hour  when  the  direction  of  the  wind  makes  an  angle  of  45° 
with  that  of  the  current? 


R  =  \/l6  -f-  49  -f  2  X  4  X  7  cos  45°  =  10.2m.     Ans. 

4.  A  w^eight  of  50  lbs.,  suspended  by  a  string,  is  drawn  aside  by  a 
horizontal  force  until  the  string  makes  an  angle  of  30°  with  the  ver- 
tical. Required  the  value  of  the  horir^ontal  force,  and  the  tension 
of  the  string.     Am.  28.8675  lbs.,  and  67.735  lbs. 

5.  Two  forces,  and  their  resultant,  are  all  equal.  What  is  the 
angle  between  the  two  forces  ?    Ans,  120°. 


Relation  between  two  Forces  and  their  Resultant. 

33.  Let  P  and  Q  be  two  forces,  and  R  their  resultant. 
Then  because  QP  is  a  parallelogram, 
the  side  PR  is  equal  to  Q.  From 
the  triangle  ORP,  because  the  sides 
are  proportional  to  the  sines  of  the 
Opposite  angles,  we  have,  ^ 

P  \  Q\  Rw^in  ORP  :  sin  ROP  :  sin  OPR. 

But,  ORP  =  QOR,  and  OPR  =  180°-^§OP;  hence,  we 
have, 
/^  :  §  :  i?  : :  sin  QOR  :  sin  ROP  :  sin  QOP ;  .  .  .  (13) 

That  is,  of  tivo  forces  and  their  resultant,  each  is  propor- 
tional to  the  sine  of  the  angJs  betioeen  the  other  ttoo. 

If  we  apply  a  force  i?'  equal  and  directly  opposed  to  R, 
the  forces  P,  Q,  and  R'  will  be  in  equilibrium.     The  aoi- 


3G 


MECHANICS. 


gles  QOR  and  QOR'  are  supple- 
ments of  each  other;  hence, 
Ein  QOR  =  sin  QOR';  tlie  angles 
ROF,  and  POR',  are  also  sup- 
plementary; hence,  sm  7?  OP  = 
sin  POR'.  We  have  also  R  =  R'. 
Making  these  substitutions  in  the 
preceding  proportion,  we  have, 

P:  Q:  R'  ::  sin  QOR'  :  sin  POR'  :  sin  QOP.    .  .  .  (14) 

Hence,  if  three  forces  are  in  equilibriu7n,  each  is  propor- 
tional to  the  sine  af  the  angle  hetiueen  the  other  two. 


Fijj.  14. 


Principle  of  Moments. 

34.  The  moment  of  a  force,  with  respect  to  a  point, 
is  the  product  of  the  intensity  of  the  force,  by  the 
perpendicular  from  the  point  to  the  direction  cf  the 
force. 

The  fixed  point  is  the  centre  of  moments ;  the  perpen- 
dicular is  the  lever  arm  of  the  force;  and  the  moment 
itself  measures  the  tendency  of  the  force  to  produce  rota- 
tion about  the  centre  of  moments. 

Let  P  and  Q  be  two  forces,  and  R  their  resultant; 
assume  a  point  C,  in  their  plane, 
as  a  centre  of  moments,  and 
from  it,  let  fall  on  the  forces, 
perpendiculars,  Cp,  Cq,  and  Cr; 
denote  these  perpendiculars  by 
p,  q,  and  r.     Then  will  Pp,  Qq,  Fig.  i5. 

and  Rr,  be  the  moments  of  P,  ft  and  R.  Draw  CO,  and 
from  P  let  fall  the  perpendicular  PS,  on  OR.  Denote  the 
angle  POP,  by  a,  ROQ,  or  its  equal,  ORP,  by  /3,  and 
BOO  by  <p. 


COMPOSITION,  ETC.,  OF   FORCES.  37 

Since  PR  =  Q,  we  have  from  the  right-angled  triangles 
OPS  and  PES,  the  equations, 

E  =  Q  cos  (3  -\-  P  cos  a. 
0  =  §  sin  ft  —  P  sin  a. 
Multiplying  botli  members  of  the  first  by  sin  cp,  and  of 
the  second  by  cos  y?  and  adding,  we  find, 

R  sin  Q?  =  Q  (sin  9  cos  (i  4-  sin  /3  cos  9)  -i- 
P  (sin  9  cos  cc  —  sin  a  cos  9). 
Whence,  by  reduction, 

E  sin  9  =  (2  sin  (9  +  i3)  -f  P  sin  (9  —  a). 
From  the  figure,  we  have, 

sin  9  =  j^,  sin   (9  -  a)=  -^,,  and  sin  (9  +  /3)  =  -^,. 

Substituting  in  the  preceding  equation,  and  reducing, 

we  have, 

Er  =  Qq  -\-  Pp. 

When  C  falls  within  the  angle  POE,  9  —  a  is  negative, 
and  the  equation  just  deduced  becomes 

Er  =  Qq  -  Pp- 
Hence,  in  all  cases,  the  moment  of  the  resultant  of  two 
forces  is  equal  to  the  algebraic  sum  of  the  moments  of  the 
forces  taken  separately. 

If  we  regard  the  force  Q  as  the  resultant  of  two  others, 
aiul  one  of  these  in  turn,  as  the  resultant  of  two  others, 
and  so  on,  the  principle  may  be  extended  to  any  number 
of  concurrent  forces  in  the  same  plane.  This  principle 
may  be  expressed  by  the  equation, 

Rr  =  I(Pp)  ....  (15) 
That  is,  the  7noment  of  the  resultant  of  any  number  of 
concurrent  forces,  in  the  same  plane,  is  equal  to  the  algebraic 
sum  of  the  moments  of  the  forces  taken  separately. 


38  MECHANICS. 

This  is  the  principle  of  moments. 

The  moment  of  the  resultant  is  the  resultant  moment; 
the  moments  of  the  components  are  component  7noments ; 
and  the  plane  passing  through  the  resultant  and  centre 
of  moments,  is  the  plane  of  mometits. 

When  a  force  tends  to  turn  its  point  of  application  about 
the  centre  of  moments,  in  the  direction  of  the  motion  of 
the  hands  of  a  watch,  its  moment  is  considered  positive; 
consequently,  when  it  tends  to  produce  rotation  in  a  con- 
trary direction,  its  moment  must  be  negative.  If  the  result- 
ant moment  is  negative,  the  tendency  of  the  system  is  to 
produce  rotation  in  a  negative  direction.  If  the  resultant 
moment  is  0,  there  is  no  tendency  to  rotation.  The  result- 
ant moment  may  become  0,  in  consequence  of  the  lever 
arm  becoming  0,  or  in  consequence  of  the  resultant  itself 
being  0.  In  the  former  case,  the  centre  of  moments  lies 
on  the  direction  of  the  resultant,  and  the  sum  of  the  mo- 
ments of  the  forces  that  tend  to  produce  rotation  in  one 
direction,  is  equal  to  that  of  those  tending  to  produce  rota- 
tion in  a  contrary  direction.  In  the  latter  case,  the  system 
is  in  equilibrium. 

»  Moment  of  a  Force  with  respect  to  an  Axis. 

35.  Let  P  be  a  force  and  OZ  any  axis.  Draw  a  line, 
AB,  perpendicular  to  the  force 
and  also  co  the  axis.  Let  A  be 
taken  as  the  point  of  applicatioii 
of  the  force,  and  at  this  point 
resolve  it  into  two  components 
F"  and  F\  the  former  parallel, 
and  the  latter  perpendicular  to 
OZ.     The   component    P"    can  Fig.ie. 

have  no  tendency  to  produce   rotation  about   the   axis; 


COMPOSITICX,  ETC.,  OF    FORCES.  39 

hence,  the  moment  of  P'  with  respect  to  B,  will  be  the 
same  as  the  moment  of  the  given  force  with  respect  to  the 
axis.  But,  P'  is  the  projection  of  P  on  a  plane  perpen- 
dicular to  the  axis,  and  B  is  the  point  in  which  this  plane 
intersects  the  axis.  Hence,  to  find  the  moment  of  a  force 
^vith  respect  to  an  axis,  jjroject  the  force  on  a  plane  perpen- 
dicular to  the  axis,  and  find  the  moment  of  the  projection 
ivith  respect  to  the  j^oint  in  which  the  perpendicular  plane 
cuts  the  axis. 

The  axis  is  an  axis  of  rotation,  and  any  plane  perpendi- 
cular to  it,  is  Q,  plane  of  rotation. 

To  find  the  resultant  moment  of  a  system  of  forces  in 
space,  with  respect  to  any  line  as  an  axis;  assume  a 
plane  perpendicular  to  the  given  line  as  a  plane  of  rota- 
tion, project  the  forces  on  it,  and  find  the  moments  of  the 
projections  with  respect  to  the  point  in  which  the  plane 
cuts  the  axis ;  these  will  be  the  component  moments. 
The  resultant  moment  is  the  algebraic  sum  of  the  com- 
ponent moments. 


Principle  of  Virtual   Moments. 

36.  Let  P  be  a  force  applied  to  the  material  point  O; 
let  0  be  moved  by  an  extraneous  force  to  some  position,  C, 
very  near  to  0;   project  the  path  OC  on 

the  direction   of   the  force :  the  projec-     p^Omp _P 

tion    Op,  or    Op',  is  called    the  virtual      f^^ 
velocity  of  the  force,  and  is  positive  when  Fig.  n. 

it  falls  on  the  direction  of  the  force,  as 
Op,  and  negative  when  it  falls  on  tlic  prolongation  of  the 
force,  as  Op'.     The  product  of  a  force  by  its  virtual  velo- 
city is  called  the  virtual  moment  of  the  force. 

Assume  the  figure  of  Article  34.     Op,  Oq,  and   Or  are 


40 


MECHANICS. 


the  virtual  velocities  of  P,  Q,  and  H, 
virtual  velocity  of  a  force  by  the 
symbol   6,  followed   by  a   small 
letter  of  the  same  name  as  that 
which  designates  the  force. 

We  have  from  the  figure,  as  in 
Article  34, 

li  ~  P  cos  a  +  Q  cos  /3. 
0  =  P  sin  a  —  §  sin  /J. 

Multiplying  both  members  of  the  first,  br  nog  9,  of  the 
second,  by  sin  9,  and  adding,  we  have, 

B  cos  ip  :=  P  (cos  a  COS  (p  +  siu  a  sin  9)   -f- 
Q  (cos  9  cos  fS  ■—  sin  9  sin  /3). 

Or,  by  reduction, 

i?  cos  9  =  P  cos  (9  —  a)  +  §  cos  (9  -f   /3). 

But,  from  the   right-angled   triangles    COp,   COq,  and 
COr,  we  have, 


(5r 


cos  9=  yyv-,,  cos  (9  —  a)  = 


6p 


and  cos  (9  +  /3)  = 


Substituting  in  the  preceding  equation,  and  reducing,  we 
have, 

\R6r  =  P^p  +  Q^q. 

Hence,  the  virtual  tnoment  of  the  resultant  of  two  forcssy 
is  equal  to  the  algebraic  sum  of  the  virtual  moments  of  the 
forces  taken  separately. 

If  we  regard  Q  as  the  resultant  of  two  forces,  and  one 
of  these  as  the  resultant  of  two  others,  and  so  on,  the 
principle  may  be  extended  to  any  number  of  forces, 
applied  at  the  same  point.  This  principle  may  be  ex- 
pressed by  the  following  equation  : 

R6r  =  ^  (P6p)  ; (16) 


COMPOSITION,  ETC.,  OF    FORCES.  41 

Hence,  the  virtual  momejit  of  the  resultant  of  any  mwi- 
her  of  concurrent  forces,  is  equal  to  the  algebraic  sum  of 
the  virtual  moments  of  the  forces  taken  separately. 

Resultant  of  Parallel  Forces. 
3T.  Let  P  and  Q  be  two  forces  in  the  same  plane,  and 
applied   at  points   invariably  con- 
nected— for  example,  at  the  points 
M  and  N  of  a  solid  body.     Their 
lines  of  direction  being  prolonged,  '^ 


will  meet  at  some  point  0;  and  if 
we  suppose  the  points  of  applica- 
tion to  be  transferred  to  0,  their 

resultant  may  be  determined  by  the  parallelogram  of  forces. 
Whether  the  forces  be  so  transferred  or  not,  the  direction 
of  the  resultant  will  always  pass  through  0,  and  will  lie 
between  P  and  Q.  Now,  supposing  the  points  of  applica- 
tion at  M  and  N,  let  the  force  Q  be  turned  about  N  as  an 
axis.  As  it  approaches  parallelism  with  P,  0  recedes 
from  M  and  N,  and  the  resultant  also  approaches  parallel- 
ism with  P.  Finally,  when  Q  becomes  parallel  to  P,  0  is 
at  an  infinite  distance  from  M  and  N,  and  the  resultant  is 
parallel  to  Panel  Q. 

Hence,  if  tioo  forces  are  parallel  and  act  in  the  same 
direction,  their  resultant  is  parallel  to  both  and  lies  between 
them.. 

Whatever  may  be  the  position  of  P  and  Q,  the  value  of 
the  resultant,  (Art.  34),  will  be  given  by  the  equation, 
7?  =  P  cos  a  +  §  cos  ^. 

But  when  the  forces  are  parallel  and  act  in  the  same 
direction,  we  have,  a  r=  0,  and  ^  =  0 ;  or,  cos  a  =  1,  and 
cos  j3  =  1.     Hence, 

Rr=.  P  ^  q.  ,  .  .  .  ,  (17) 


4:2  MECHAN^ICS. 

That  is,  (he  intensity  of  the  resultant  is  equal  to  the  sum  of 
the  i7ite7isities  of  the  two  forces. 

Let  Pand  Q  be  parallel  forces  acting  in  the  same  direc- 
tion, R  their  resultant,  and  S 

the  point  in  which  the  direc-  is^ ,,^0 

tion  of  R  cuts  the  line  ioininpf  /  ir. 

the  points  of  application   of         , '        j  _^ 


P  and  Q.     Through  N  draw  I 

NL,    perpendicular     to     the  Fi^.  20. 

forces,  and  take  (!,  its  inter- 
section Avith    R,  as  a  centre  of  moments.     The  centre  of 
moments  being  on  tlie  direction  of  the  resultant,  the  lever 
arm  of  the  resultant  will  be  0,  and  from  the  principle  of 
moments,  (Art.  34),  we  have, 

r  X  CL  =:  Q  X  ON; 
or,  R:  Q::  ON :  CL. 

But,  from  the  similar  triangles  (7iV^/Vand  LNM,  we  have, 

CN'.  CL'.'.  8N'.  SM. 
Combining  the  two  proportions,  we  have, 
P:  Q::  SN:  SM. (18) 

That  is,  t?ie  resultant  divides  the  line  joining  the  poinfsi 
of  application  of  the  compo7ie7ifs,  imwrsely  as  the  co  i/po- 
ne?its. 

If  a  force  R'  be  applied  at  S  equal  and  directly  opposed 
to  R,  it  will  hold  F  and  Q  in  equilibrium.  The  forces  R', 
P,  and  Q,  being  in  equilibrium,  Q  must  be  equal  and 
directly  opposed  to  the  resultant  of  R'  and  P.  But,  R' 
and  P  are  parallel  and  act  in  opposite  directions,  R'  being 
the  greater.  Hence,  the  resultant  of  two  parallel  forces 
acting  in  opposite  directions,  is  parallel  to  hotli,  lies  without 
both,  on  the  side  and  in  the  direction  of  the  greater,  and  its 


COMPOSITION,  ETC.,  OF   FORCES.  43 

intensity  is  equal  to  the  difference  of  the  intensities  of  the 
given  forces. 

If  P  and  Q,  Fig.  21,  repre-  y 

sent  two  sucli    forces,  and  R 


their    resultant ;    it    may    be  /      i'^ 

shown,   as    in    the    preceding         g^^^ i — 5.-B, 

article,  that.  Fig.  21. 

P:  Q::;SJV:SM.....(19) 

By  composition,  we  find, 

P:  Q:  P  -{-  Q::  SjV  :  SM:  SJV  4-  SM; 
and  by  division, 

P:  Q:  P  -  Q::  SN:  SM :  SN -  SM. 

When  the  forces  act  in  the  same  direction,  as  in  Fig.  20, 
P  +  Q=  R,  and  SN  +  .S'.^  =  MN. 

When  they  act  in  opposite  directions,  as  in  Fig.  21, 
p  -  Q  =  R^  and  SN  -  SM  =  MN. 

Substituting  in  the  preceding  proportions,  for  P  -\-  Q, 
P  —  Q,  SN  +  SM,  and  SN  —  SM,  their  values,  w^e  have, 
P:  Q:  R::  SN:  SM :  MN. (20) 

That  is,  of  any  two  parallel  forces  and  their  resultant, 
each  is  proportional  to  the  distance  betiveen  the  other  two. 

We  see,  from  the  preceding  proportion,  that  so  long  as 
the  intensities  of  P  and  Q  and  their  points  of  applica- 
tion remain  unchanged,  the  values  of  SM  and  SN  also 
remain  unchanged,  no  matter  what  direction  the  forces 
may  have.  Hence,  if  two  parallel  forces  be  turned  about 
their  points  of  application,  their  intensities  remaining 
unchanged,  their  resultant  will  turn  about  a  fixed  point 
and  continue  parallel  to  the  given  forces.  This  fixed  point 
is  called  the  centre  of  the  parallel  forces. 

If  P  and  Q  be  equal  and  act  in  opposite  directions,  R 
will  be  0,  and  *S'  will  be  at  an  infinite  distance.     Two  such 


44 


MECHANICS. 


forces  constitute  a  couple.  The  tendency  of  a  couple  is  to 
produce  rotation ;  the  measure  of  this  tendency,  called  the 
moment  of  the  couple^  is  the  product  of  one  of  the  forces, 
by  the  distance  between  the  twa 


M' 


«T 


Geometrical  Composition  and  Resolution  of  Parallel  Forces. 

38.  The  preceding  principles  give    the   following  geo- 
metrical constructions.  .p. 

1.  To  find  the  resultant  of  two 
parallel  forces  lying  in  the  same  direc- 
tion: 

Let  P  and  §  be  the  forces,  M  and  N     ""'     /Ts 
their     points     of     application.       Make 
MQ'  =  Q,  and  JVF'  =  P;    draw  F'  Q',       *  0 

cutting  ^¥JV"  in  S;  through  S  draw  SR 
parallel  to  MP,  and  make  it  equal  to 
P  -\-  Q  J  it  will  be  the  resultant. 

For,  from  the  triangles  P'S^  and 
Q'SM,  we  have, 

P'N:  Q'M:  :  SI^:  8M;  or,  P :  Q  : 

2.  To  find  the  resultant  of  two  parallel  forces  acting  in 
opposite  directions : 

:     Let  P  and  Q  be  the  forces,  M  and  N 
their  points  of  application.     Prolong  QN' 
till  NA  =  P,  and  make  3IB  ==  Q;  draw 
AB,  and  produce  it  till  it  cuts  NM  pro- 
duced in  S;  draw  SP  parallel  to  MP,  and        [        b 
equal  to  BP,  it  will  be  the  resultant  re- 
quired. 

For,  from  the  triangles  SNA  and  SMB,  pjg. »? 

we  have, 

AN'  BM:  :  SN :  SM ;  or,  P :  Q  :  :  SN:  SM. 


B 

Fig.  22. 

SN :  SM. 


M 

-i 


/B 


A^ 


COMPOSITION,  ETC.,  OF   FORCES. 


45 


Mc- 


M^' 


Fig.  24. 


-;.-N 


;q 


3.  To  resolve  a  force   into  two  parallel  components  in 
the  same  direction,  and  applied  at  given  points : 

Let  E  be  the  force,  M  and  JV  the 
points  of  application.  Through  M  and 
/V  draw  lines  parallel  to  E.  Make 
MA  =  E,  and  draw  AN,  cutting  R  in 
B;  make  MP  =  SB  and  NQ  =  BE; 
they  will  be  the  components. 

For,  from  the  triangles  AMN  and 
BSN, 

BS:  AM'.'.  SN:  MN; 
or,  ^.S':    E   \:SN:MN, 

But,  from  proportion  (20),  we  have, 

P'.E'.'.SN.MN; 
.-.  BS  =  P,  and  BE  =   Q. 

4.  To   find   the   resultant  of    any   number   of    parallel 
forces. 

Let  P,  P\  P",  P'",  be  parallel  forces, 
ant  of  P  and  P,  by  the  rule  already 
given,  it  will  be  i?'  =  P  +  P' ;    find 
the  resultant  of  E'  and  P",  it  will  be 
B"  =  P  ^  P'  +  P" ;  find  the  result-     f 
ant  of  R"  and  P"\  it  will  be  7?  =  P  +     p 
P'  +  P"  +  P".     If  there  be  a  greater 
number  of  forces,  the  operation  of  com- 
position may  be  continued;  the  final 
result  will  be  the  resultant  of  the  sys- 
tem.    If  some  of  the  forces  act  in  con- 
trary directions,  combine  all  that  act  in  one  direction,  as 
just  explained,  and  call  their  resultant  R' ;  then  combine 
all  that  act  in  a  contrarv  direction,  and  call  their  resultant 


Find  the  result- 


-llf  I  i 


B" 


pn 


E' 


R 
Fiff.  25. 


46  MECHANICS. 

B";  finally,  combine  R'  and  R";  their  resultant,  i?,  will 
be  the  resultant  of  the  system. 

If  the  forces  P,  P \  &c.,  be  turned  about  their  points  of 
application,  their  intensities  remaining  unchanged,  the 
forces  R',  R",  R,  will  also  turn  about  fixed  points,  contin- 
uing parallel  to  the  given  forces.  The  point  through  which 
R  always  passes,  is  called  the  centre  of  parallel  forces. 

Oo-ordinates  of  the  Centre  of  Parallel  Forces. 

39.  Let  P,  P',  P",  &c.,  be  paiallel  forces,  applied  at 
points  that  maintain  fixed  positions  with  respect  to  a  sys- 
tem of  rectangular  axes,  and  let  R,  equal  to  -  (P),  be  their 
resultant.  Denote  the  co-ordinates  of  the  points  of  appli- 
cation of  the  forces  by  x,  y,  z  ;  x,  y',  z',  &c. ;  and  those  of 

^  by  ^. '  2//  J  ^'  • 

Turn  the  forces  about   their  points  of  application,  till 

they  are  parallel  to  the  axis  of  Y,  and  in  that  position  find 

their   moments  with  respect   to   the  axis  of  Z.     In    this 

position  the  lever  arms  of  the  forces  are  x,  x',  &c.,  and  the 

lever  arm  of  R  \^  x^.     From  the  principle  of  moments, 

(Art.  34),  we  have 

Rx,  =  Px  +  P"x'  +,  &c. 
or,  ■ 

^/  =  ~~lTP) ^^^^ 

By  making  the  forces  in  like  manner  parallel  to  the  axis 
of  Z,  and  taking  their  moments  with  respect  to  the  axis 
of  X,  we  luive, 

.v.=  -;^f (^2) 

And  in  like  manner,  we  find, 


(23) 


COMPOSITION',  ETC.,  OF    FORCES.  47 

From  either  of  the  above  expressions  we  infer  that  the 
lever  arm  of  the  residtant  of  a  system  of  imrallel  forces 
ivith  respect  to  any  axis  per2)endicular  to  the  forces,  is  equal 
to  the  algebraic  sum  of  the  moments  of  the  forces  with  re- 
spect to  that  axis,  divided  hy  the  algebraic  sum  of  the  forces. 

Equations  21,  22,  and  23,  determine  the  position  of  the 
centre  of  parallel  of  forces. 

Composition  of  Forces  in  Space,  applied  at  points  invariably 
connected. 

40.  Let  P,  P',  P",  &c.,  be  forces  in  space,  applied  at 
points  of  a  solid  body.     Assume 
a  point  0,  and  through  it  draw  S 

three   perpendicular  axes.     De-  *^ 

note  the  angles  that  P,  P',  P", 

&c.,  make  with  the  axis  of  X, 

by   a,   a',    a.",  &c. ;    the  angles 

they  make  with  the  axis  of  Y, 

by  ft  d',  13",    &c.;    the  angles.    :^- — ^--y^ 

they  make  with  the  axis  of  Z,  Fig.  26. 

^y  7>  r'j  7">  &c.,  and  denote  the  co-ordinates  of  their  points 

of  application  by  x,  y,  zj  x',  y,  z' ;  x",  y",  z",  &c. 

Let  each  force  be  resolved  into  components  parallel  to 
the  axes. 

"We  have  for  the  group  parallel  to  the  axis  of  JT, 
Pcosa,  P'cosa',  P"cosx",  &c. ; 
for  the  group  parallel  to  the  axis  of  Y, 

Pcosi3,  P'cos/3',  P"cos'3",  &c.; 
and,  for  the  group  parallel  to  the  axis  of  Z, 
Pcosy,  P'cosy,  P"cosy",  &c. 

Denoting  the  resultants  of  these  several  groups  by  JT,  Y, 
and  Z,  we  have. 


^ 


-f^^COStt 


48  MECHANICS. 

X=z  I{Pcom),Y=  I{Fcos(3),  and  Z=  I(Pcosy).  .  .  .  (24) 

If  the  given  forces  have  a  single  resultant,  the  forces 
JT,  Y,  and  Z,  will  be  applied  at  a  point,  the  co-ordinates  of 
which  are  the  same  as  the  lever  arms  of  the  forces,  each 
taken  with  respect  to  the  axis  whose  name  comes  next  in 
order.  Denoting  these  co-ordinates  by  x^,  y^,  and  z^,  we  have, 
as  in  Art.  39, 

2*(Pcos/3  x)  ^ 


y. 


2'(Pcos7  y) 
liPcosy) 


(25) 


_  2'(Pcosa  z) 
^'  ~     2'(Pcosa) 

These  determine  the  point  of  application  of  the  result- 
ant. Denoting  the  resultant  by  R,  and  the  angles  it  makes 
with  the  axes  by  a,  h,  and  c,  we  have,  from  preceding  prin- 
ciples, 

R  =  Vx^  +  Y'  +Z^ (26) 

and 

X  Y  Z 

cos  a  —   ^,  cos  Z>  =  -75,  cos  c  =  ^  .  .  .  .  (27) 
Ji  Ji  K 

Hence,  the  resultant  is  completely  determined. 

If  the  forces  are  in  a  plane,  that  plane  may  be  taken  as 
the  plane  XY.  In  this  case  the  formulas  for  determining 
the  point  of  application  of  the  resultant  become, 

'         2(i^cos<:^)  •  '         2(Pcosa) 

and  the  formulas  for  finding  the  intensity  and  direction  of 
the  resultant  reduce  to, 

R  =  ^Xn^T',  cos  «  =  ^,  cos  Z>  =  -^,  .  .  .  .(29) 

Ji  Ji 

in  which 

X=  H  (Pcosa)  and  F  =  J  (Pcosi3)  .... 


COMPOSITION,  ETC.,  OF   FORCES.  49 

Conditions  of  Equilibrium. 

41.  A  system  of  forces  applied  at  points  of  a  solid  body 
will  be  in  equilibrium  when  they  have  no  tendency  to  pro- 
duce motion,  either  of  translation,  or  of  rotation.  We 
have  seen  that  any  system  of  forces  can  be  resolved  into 
three  groups,  parallel  to  three  rectangular  axes.  The  tend- 
ency of  each  group  is  to  produce  motion  parallel  to  the 
corresponding  axis,  and  the  tendency  of  the  groups  taken 
two  and  two  is  to  produce  rotation  about  the  axis  to  which 
they  are  both  perpendicular.  In  order  that  there  may  be 
no  tendency  to  either  kind  of  motion,  we  must  have  the 
following  relations,  called  conditions  of  equilibrium  : 

1st.  TJie  algebraic  sum  of  the  components  of  the  forces  in 
the  direction  of  any  three  rectangular  axes  must  be  separately 
equal  to  0. 

2d.  Tlie  algebraic  sum  of  the  moments  of  the  forces,  with 
respect  to  any  three  rectangular  axes,  must  be  separately 
equal  to  0. 

If  the  forces  lie  in  a  plane,  the  conditions  of  equilibrium 
reduce  to  these :  • 

1st.  The  algebraic  sum  of  the  components  of  the  forces,  in 
the  directio7i  of  any  two  rectangular  axes,  separately  equal 
too, 

2d.  Tlie  algebraic  sum  of  the  moments  of  the  forces,  with 
respect  to  any  point  in  the  plane,  equal  to  0. 

If  a  body  is  restrained  by  a  fixed  axis,  as  in  case  of  a 
pulley,  or  wheel  and  axle,  the  forces  will  be  in  equilibrium 
when  the  algebraic  sum  of  the  moments  of  the  forces  with 
respect  to  the  axis  is  equal  to  0. 

This  case  is  one  frequently  met  with  in  discussing 
machines 


CHAPTER  III. 

CIlinilE   OF   GKAVITY   Al^D   STABILITY. 
Weight. 

42.  The  force  of  gravity  acts  on  all  the  particles  of  a 
body,  tending  to  draw  them  toward,  the  centre  of  the 
earth.  If  this  force  be  resisted  it  produces  a  pressure 
called  weight.  The  weight  of  a  body  is  the  resultant  of 
the  weights  of  all  its  particles.  The  weights  of  the  parti- 
cles are  sensibly  directed  toward  the  centre  of  the  earth, 
and  if  the  body  be  small  in  comparison  with  the  earth,  they 
may  be  regarded  as  parallel  forces ;  hence,  the  weight  of  a 
body  is  parallel  to  the  weights  of  its  particles,  and  is  equal 
to  their  sum. 

Centre  of  Gravity. 

43.  The  centre  Sf  gravity  of  a  body  is  the  point  through 
which  the  direction  of  its  weight  always  passes.  The 
weight  being  the  resultant  of  parallel  forces,  the  centre  of 
gravity  is  a  centre  of  parallel  forces,  and  so  long  as  the 
relative  position  of  the  particles  remains  unchanged,  this 
point  retains  a  fixed  position  in  the  body.  The  position  of 
the  centre  of  gravity  is  entirely  independent  of  the  value 
of  the  intensity  of  gravity,  provided  we  regard  this  force  as 
constant  throughout  the  body,  which  we  may  do  in  most 
cases.  Hence,  the  centre  of  gravity  is  the  same  for  the 
same  body,  wherever  it  may  be  situated.  The  determina- 
tion of  the  centre  of  gravity  is,  then,  reduced  to  the  deter- 
mination of  the  centre  of  a  system  of  parallel  forces. 


CENTRE    OF   GRAVITY    AND    STABILITY.  51 

In  what  follows,  the  lines  and  snrfaces  treated  of,  are 
regarded  as  material,  that  is,  made  np  of  inaterial  poi7its. 
The  volumes  considered,  are  supposed  to  be  homogeneous, 
so  that  the  weights  of  different  parts  are  proportional  to 
their  volumes.  This  supposition  reduces  the  operation  of 
finding  the  centre  of  gravity,  to  the  geometric  one  of  find- 
ing the  centre  of  figure. 

Preliminary  Principles. 

44.  Let  there  be  any  number  of  parallel  forces  applied 
at  points  of  a  straight  line.  If  we  apply  the  method  of 
finding  the  point  of  apjMcation  of  their  resultant,  as  ex- 
plained in  Art.  39,  it  will  be  seen  that  it  lies  on  the  given 
line.  Hence,  the  centre  of  gravity  of  a  material  straight 
line  is  on  that  line. 

In  like  manner  it  may  be  shown  that  the  centre  of  gravity 
of  a  plane  curve,  or  of  a  plane  area,  is  in  that  plane. 


i\o.  -uet  m  ana  ^v  oe  two  material  pointa,      ^. 
equal  in  weight,  and  firmly  connected  by  a      | 
line  MN.     The  resultant  of  these  weights     \ 


« 


B 


Centre  of  Gravity  of  a  Straight  Line. 
45.  Let  J/ and  A^  be  two  material  points,      ^ ^ 

I 

will  bisect  the  line  MN  in  S  (Art.  37) ; 
fience,  S  is  the  centre  of  gravity  of  M 
and  K  Pi^^  2^_ 

Let  MN  be  a  material  straight  line,  and  S  its  middle 
point.     We  may  regard  it  as  composed  of  material  points 
A,  A';  B,  B',  &c.,  equal  in  weight,  and 
symmetrically  disposed  with  respect  to      ab         S        B'-AL 
S.    From  what  precedes,  the  centre  of 
gravity    of    each    pair    of  equidistant 
points  is  at  S;  consequently  the  centre  f 

of  gravity  of  the   whole  line  is  at  S.  Fig  28 


52  MECHAlflCS. 

That  is,  the  centre  of  gravity  of  a  straight  liiie  is  at  its  mid- 
dle point. 

Additional  Principles. 

46.  A  line  of  symmetry  of  a  plane  figure  is  a  straight 
line  that  bisects  a  system  of  parallel  chords  of  the  figure. 
If  the  line  is  perpendicular  to  the  chords  it  bisects,  it  is  a 
line  of  right  symmetry,  otherwise  it  is  a  line  of  ohliqiie  sym- 
metry.  The  axes  of  an  ellipse  are  lines  of  right  symmetry ; 
other  diameters  are  lines  of  oblique  symmetry. 

A  jjlane  of  symmetry  of  a  surface,  or  volume,  is  a  plane 
that  bisects  a  system  of  parallel  chords  of  the  figure.  It 
may  be  a  plane  of  right,  or  a  plane  of  obliqne  symmetry. 

The  intersection  of  two  planes  of  symmetry  is  an  axis 
of  symmetry. 

Let  A  QBP  be  a  curve,  and  AB  o.  line  of  symmetry, 
bisecting  the  parallel  chords  PQ.     The  centre  of  gravity 
of  each  pair  of  points  P,  ft  is  on  AB 
(Art.  45),  hence,  the  centre  of  gravity 
of  the  entire  curve  is  on  AB  (Art.  44). 
Again,  the  centre  of  gravity  of  each 
chord  PQ  is  on  AB,  hence  the  centre 
of  gravity  of  the   entire   area  is  on 
AB.     That  is,  if  a  i^lane  curve,  or  a 
plane  area,  has  a  line  of  symmetry,  its  centre  of  gravity  is 
on  that  line. 

In  like  manner,  if  a  surface  or  volume  has  a  plane  of 
symmetry,  its  centre  of  gravity  is  in  that  plane. 

Two  lines  of  symmetry,  or  three  planes  of  symmetry 
intersecting  in  a  point,  are  sufficient  to  determine  the  cen- 
tre of  gravity  of  the  corresponding  magnitude.  Thus,  all 
diameters  of  the  circle  are  lines  of  symmetry,  and  because 
they  intersect  at  the  centre,  it  follows  that  the  centre  of 


CENTRE    OF   GRAVITY    AND    STABILITY.  53 

gravity  of  both  the  circumference  and  area,  is  at  the  centre. 
For  a  similar  reason  the  centre  of  gravity  of  both  circum- 
ference and  area  of  an  ellipse  is  at  its  centre. 

Any  plane  through  the  centre  of  a  sphere,  or  of  an 
ellipsoid,  is  a  plane  of  symmetry;  hence  the  centre  of 
gravity  of  either  is  at  its  centre. 

The  centre  of  gravity  of  any  surface  or  volume  of  revo- 
lution is  on  its  axis. 

Centre  of  Gravity  of  a  Triangle. 

47.  Let  ABC  be  a  plane  triangle.  Join  the  vertex  A 
v^^ith  the  middle  point  D,  of  the  opposite  side  BC;  then 
will  AD  bisect  all  lines  drawn  in  the 

triangle  parallel  to  BC,  it  is  therefore  A 

a  line  of  symmetry :  hence,  the  centre  /t\ 

of  gravity   of  the   triangle  is   on  AD  yy    j    \ 

(Art.  46) ;  for  a  like  reason  it  is  on  BE,  /  \  "f-.^    \ 

drawn  from  the  vertex  B  to  the  middle  ^ /i-.:::::.^^z-.-.zr:^i\^ 
of  the  side  A  C ;  it  is,  therefore,  at  G,  Fig.  so. 

their  point  of  intersection. 

DrawjB'Z);  then,  since  ED  bisects  AC  and  BC,  it  is 
parallel  to  AB,  and  the  triangles  EGD  and  AGB  are  sim- 
ilar. The  side  ED  is  one-half  its  homologous  side  AB, 
consequently  the  side  GD  is  one-half  its  homologous  side 
AG;  that  is,  G  is  one-third  of  the  distance  from  D 
to  A. 

Hence,  the  centre  of  gravity  of  a  plane  triangle  is  on  a 
line  draion  from  the  vertex  to  the  middle  of  the  base,  and  at 
one- third  the  distance  from  the  base  to  the  vertex. 

Centre  of  Gravity  of  a  Parallelogram. 

48.  Let  ^ICbea  parallelogram.  Draw  ^5'/^  bisecting  AB 
and    CD;    it   will   bisect   all   lines   of   the   parallelogram 


64  MECHANICS. 

parallel  to  these;  hence,  the  centre  of  gravity  is  on  it; 
draw  also  OH  bisecting  AD  and  BC ;  for  a  similar  rea- 
son, the  centre  of  gravity  is  on  it ;  it  is,         d        e         C 
therefore,  at  G,  their  point  of  intersec-       /        /         / 
tion.  f f---k 

Hence,  the  centre  of  gravity  of  a  parol-  ^ ^ ^ 

lelogram    is  at   the  intersection  of   tivo  ^Jg-  3i. 

straight  lines  joining  the  middle  points  of  the  opposite  sides. 

The  diagonals  of  a  parallelogram  are  also  linos  of  sym- 
metry, each  bisecting  the  chords  parallel  to  the  other. 
Hence,  the  centre  of  gravity  is  at  their  intersection. 

Centre  of  Gravity  of  a  Trapezoid. 

49.  Let  ^  (7  be  a  trapezoid.  Join  the  middle  points,  0 
and  P,  of  the  parallel  sides,  by  a  straight  line ;  this  will 
bisect  all  lines  parallel  to  DC ;  hence, 

it  must  contain  the  centre  of  gravity. 
Draw  BD,  dividing  the  trapezoid  into 
two  triangles.  Draw  also  DO  and 
BP;  take  OQ=\OD,  and  Pi?  = 
\PB ;  then  will  Q  and  R  be  the  cen- 
tres of  gravity  of  these  triangles  (Art.  47).  Join  Q  and  R 
by  a  straight  line ;  the  centre  of  gravity  of  the  trapezoid 
must  be  on  this  line  (Art.  44).  Hence,  it  is  at  G  where 
QR  cuts  OP. 

Centre  of  Gravity  of  a  Polygon. 

50.  Let  ABODE  bo  a  polygon,  and 
a,  hj  c,  d,  G,  the  middle  points  of  its 
sides.  The  weights  of  the  sides  are  pro- 
portional to  their  lengths,  and  may  be 
represented  by  them. 

Let  it  be  required  to  find  the  centre 


CENTRE   OF   GRAVITY    AND    STABILITY.  55 

of  gravity  of  the  perimeter  ;  join  a  and  h,  and  find  a  point 

0,  such  that 

ao  :  ob  ::  BC  :  BA; 

then  will  o  be  the  centre  of  gravity  of  AB  and  BC. 
Join  0  and  c,  and  find  a  point  o',  such  that 

oo'  :  o'c  :  :  CD  :  AB  +  BC; 

then  willo'  be  the  centre  of  gravity  of  the  three  sides,  AB, 
BC,  and  CD.  Join  o'  with  d,  and  proceed  as  before,  con- 
tinuing the  operation  till  the  last  point,  G,  is  found  ;  this 
v/ill  be  the  centre  of  gravity  of  the  perimeter. 

To  find  the  centre  of  gravity  of  the  area,  divide  it  into 
triangles,  and  find  the  centre  of  gravity  of  each  triangle. 
The  weights  of  these  triangles  are  pro- 
portional to  their  areas,  and  may  be 
represer^d  by  them.  Let  0,  0',  0", 
be  the  centres  of  gravity  of  the  trian- 
gles into  which  the  polygon  is  divided. 
Join  0  and  0',  and  find  a  point  0'", 
such  that  Fig.  34. 

O'O'"  :  00'"  ::  ABC  :  ACDj 

then  will  0'"  be  the  centre  of  gravity  of  the  triangles  ABC 
and  A  CD. 

Join  0"  and  0'",  and  find  a  point,  G,  such  that 

0"'G  :  0"G  :  :  ADB  :  ABC  +  ACD j 

then  will  G  be  the  centre  of  gravity  of  the  polygon. 

To  find  the  centre  of  gravity  of  a  curvilinear  area  by 
approximation,  we  draw  a  polygon  whose  perimeter  shall 
nearly  coincide  with  that  of  the  given  area,  and  then  find 
its  centre  of  gravity.  The  accuracy  of  this  method  will 
depend  on  the  closeness  with  which  the  polygon  approaches 
the  curvilineal  area. 


56  MECHANICS. 

Centre  of  Gravity  of  a   Pyramid. 

51.  A  pyramid  may  be  regarded  as  made  up  of  infinitely 
thin  layers,  parallel  to  either  of  its  faces.  If  a  line  be 
drawn  from  either  vertex  to  the  centre  of  gravity  of  the 
opposite  face,  it  will  pass  through  the  centres  of  gravity 
of  ail  the  layers  parallel  to  that  face.  We  may  consider 
the  weight  of  each  layer  as  applied  at  its  centre  of  gravity, 
that  is,  at  a  point  of  this  line ;  hence,  the  centre  of  gravity 
of  the  pyramid  is  on  this  line,  (Art.  44). 

Let  A  BCD  be  a  triangular  pyramid,  and  K  the  middle 
point  of  DC.  Draw  KB  and  KA  ;  lay  off  KO  =  ^KB, 
and  KO'  =  ^KA.  Then  will  0  be  the 
centre  of  gravity  of  the  face  DBC,  and 
0'  that  of  the  face  CAD.  Draw  AG 
and  BO'  intersecting  in  G.  Because 
the  centre  of  gravity  of  the  pyramid  is 
on  both  AO  and  BO',  it  is  at  their 
intersection  G.  Draw  00' ;  then  KO 
and  KO'  being  third  parts  of  KB  and 
KA,  00'  is  parallel  to  AB,  and  the 
triangles  OGO'  and  AGB  are  similar,  consequently  their 
homologous  sides  are  proportional.  But  00'  is  one-third 
o{  AB,  OG  is  therefore  one-third  of  GA,  or  one-fourth 
of^O. 

Hence,  the  centre  of  gravity  of  a  triaiigular  pyramid  i$ 
on  a  line  drawn  from  its  vertex  to  the  centre  of  gravity  of 
its  hase,  and  at  one-fourth  the  distance  from  the  lase  to 
the  vertex. 

Either  face  of  a  triangular  pyramid  may  be  taken  as  tlie 
base,  the  opposite  vertex  being  the  vertex  of  the  pyramid. 

To  find  the  centre  of  gravity  of  a  polygonal  pyramid 
A'BCDEF,  A  being  the  vertex.     Conceive  it  divided  into 


CENTRE    OF    GRAVITY    AND    STABILITY.  57 

triangular  pyramids,    having  a  common 

vertex    A,      If  an    auxiliary    plane    be 

passed  parallel  to  the  base,  at  one-fourth 

of  the   distance   from   the   base   to    the 

vertex,   it  follows,   from  what   has  just 

been  shown,  that  the  centres  of  gravity 

of  all  the  partial  pyramids  will  lie  in  this 

plane ;  hence,   the  centre  of  gravity  of 

the  entire  pyramid  must  lie  in  this  plane  (Art.  44).     But 

it  has  been  shown,  that  the  centre  of  gravity  is  somewhere 

on  the  line  drawn  from  the  vertex  to  tlie  centre  of  gravity 

of  the  base ;  it  must,  therefore,  be  where  this  line  pierces 

the  auxiliary  plane  : 

Hence,  the  centre  of  gravity  of  any  loyramid  is  on  a  line 

drawn  from  its  vertex  to  the  centre  of  gravity  of  its  hase, 

and  at  one  fourth  the  distance  from  the  hase  to  the  vertex. 
A  cone  is  a  pyramid  having  an  infinite  number  of  faces : 
Hence,  the  centre  of  gravity  of  a  cone  is  on  a  line  drawn 

from  the  vertex  to  the  centre  of  gravity  of  the  hase,  arid  at 

one  fourth  the  distance  from  the  hase  to  the  vertex. 

Centre   of  Gravity  of  a  Prism. 

52.  A  prism  is  made  up  of  layers  parallel  to  the  bases, 
and  if  a  straight  line  be  draAvn  between  the  centres  of 
gravity  of  the  bases  it  will  pass  through  the  centres  of 
gravity  of  all  the  layers  ;  the  centre  of  gravity  of  the 
prism  is,  therefore,  on  this  line,  which  Ave  may  call  the 
axis  of  the  prism.  The  prism  is  also  made  up  of  filaments, 
parallel  to  the  lateral  edges,  and  if  a  plane  be  passed  paral- 
lel to  the  bases  of  the  prism  and  midway  between  them,  it 
will  contain  the  centres  of  gravity  of  all  the  filaments ; 
the   centre  of  gravity  of  the   prism  is  therefore  in   thia 

plane.     It  must  then  be  where  this  piano  cuts  the  axis  - 

3* 


58  MECHAKICS. 

Hence,  the  centre  of  gravity  of  a  2^ris7}i  is  at  the  middle 
of  its  axis. 

A  cylinder,  is  a  prism  whose  bases  have  an  infinite  num- 
ber of  sides : 

Hence,  the  centre  of  gravity  of  a  cylinder  luhose  bases  are 
parallel  is  at  the  niiddle  of  its  axis. 

Centre  of  Gravity  of  a   Polyhedron. 

53.  If  a  point  within  a  polyhedron  be  joined  with  eacli 
vertex  of  the  polyhedron,  we  shall  form  as  many  pyramids 
as  the  solid  has  faces :  the  centre  of  gravity  of  each  pyra- 
mid may  be  found  by  the  rule.  If  the  centres  of  gravity 
of  the  first  and  second  pyramid  be  joined  by  a  straight 
line,  the  common  centre  of  gravity  of  the  two  may  be 
found  by  a  process  similar  to  that  used  in  finding  the  cen- 
tre of  gravity  of  a  polygon,  observing  that  the  weights  of 
the  pyramids  are  proportional  to  their  volumes,  and  may 
be  represented  by  them.  Having  compounded  the  weights 
of  the  first  and  second,  and  found  its  point  of  applica- 
tion, we  may,  in  like  manner,  compound  the  weight  of 
these  two  with  that  of  the  third,  and  so  on  ;  the  last  point 
of  application  will  be  the  centre  of  gravity  of  the  poly- 
hedron. 

The  centre  of  gravity  of  a  body  bounded  by  a  curved 
surface  may  be  found  by  approximation,  as  follows :  Con- 
struct a  polyhedron  whose  faces  are  nearly  coincident  with 
the  surface  of  the  given  body  and  find  its  centre  of  gravity 
by  the  method  just  explained ;  this  will  be  the  point 
sought. 

The  accuracy  of  the  method  will  depend  upon  tlie  clost- 
ness  between  the  given  figure  and  the  polyhedron.  . 

The  methods  of  finding  the  centre  of  gravity,  already 
given,  are  sufficient  for  must  purposes.     Tlie  most  general 


CENTRE   OF   GRAVITY    AND   STABILITY.  59 

method,  however,  depends  on  the  Differential  and  Integral 
Calculus. 

Experimental  determination  of  the  Centre  of  Gravity. 

54.  The  weight  of  a  body  always  passes  through  its 
centre  of  gravity,  no  matter  Avhat  may  be  the  position  of 
the  body.  If  we  attach  a  flexible  cord  to  a  body  at  any 
point  and  suspend  it  freely,  it  must  ultimately  come  to  a 
state  of  rest.  In  this  position,  the  body  is  acted  upon  by 
two  forces:  its  weight,  tending  to  draw  it  toward  the  cen- 
tre of  the  earth,  and  the  tension  of  the  cord,  that  resists 
this  force.  In  order  that  the  body  may  be  in  equilibrium, 
these  forces  must  be  equal  and  directly  opposed.  But  the 
direction  of  the  weight  passes  through  the  centre  of  gravity 
of  the  body ;  hence,  the  tension  of  the  string,  which  acts 
in  the  direction  of  the  string,  must  also  pass  through  the 
same  point.  This  principle  gives  rise  to  the  following 
method  of  finding  the  centre  of  gravity  : 

Let  ABC  be  a  body  of  any  form  whatever.  Attach  a 
string  to  any  point,  C,  and  suspend  it  freely;  when  the 
body  comes  to  rest,  mark  the  direction 
of  the  string ;  then  suspend  the  body  by 
a  second  point,  B,  and  when  it  comes 
to  rest,  mark  the  direction  of  the 
string ;  the  point  of  intersection,  G, 
will  be  the  centre  of  gravity  of  the 
body.  Fig.  37. 

Instead  of  suspending  the  body  by  a  string,  it  may  be 
balanced  on  a  point.  In  this  case,  the  weight  acts  verti- 
cally downward,  and  is  resisted  by  the  reaction  of  the 
point ;  hence,  the  centre  of  gravity  lies  vertically  over  the 
point. 

If,  therefore,  a  body  be  balanced  at  any  two  points  of  its 


60  MECHANICS. 

surface,  and  verticals  be  drawn  through  the  points,  in  these 
positions,  their  intersection  will  be  the  centre  of  gravity  of 
the  body. 

If  a  body  be  suspended  by  an  axis,  it  can  only  be  at  rest 
when  the  centre  of  gravity  is  in  a  vertical  plane  through 
the  axis. 

The  centre  of  gravity  may  be  above,  below,  or  on  the 
axis. 

In  the  first  case,  if  the  body  be  slightly  deranged,  it  will 
continue  to  revolve  till  the  centre  of  gravity  falls  below  the 
axis ;  in  the  second  case,  it  will  return  to  its  primitive  posi- 
tion ;  in  the  third  case,  it  will  remain  in  the  position  in 
which  it  is  placed. 

Centre  of  Gravity   of  a  System  of  Bodies. 

55.  When  we  have  several  bodies,  and  it  is  required  to 
find  their  common  centre  of  gravity,  it  Avill  often  be  found 
convenient  to  employ  the  principle  of  moments.  To  do  this, 
we  first  find  the  centre  of  gravity  of  each  body  sei)arately, 
by  rules  already  given.  The  weight  of  each  body  is  then 
regarded  as  a  force,  applied  at  the  centre  of  gravity  of  the 
body.  The  weights  being  parallel,  we  have  a  system  of 
parallel  forces,  whose  points  of  application  are  known.  If 
these  points  are  all  in  the  same  plane,  we  find  the  lever 
arms  of  the  resultant  of  all  the  weights,  with  respect  to 
two  lines,  at  riglit  angles  to  each  other  in  that  plane;  and 
these  will  make  known  the  point  of  application  of  the  re- 
sultant, or,  what  is  the  same  thing,  the  centre  of  gravity  of 
the  system.  If  the  points  are  not  in  the  same  plane,  the 
lever  arms  of  the  resultant  are  found,  with  respect  to  three 
axes,  at  right  angles  to  each  other ;  these  make  known  the 
point  of  application  of  the  resultant  weight,  or  the  position 
of  the  centre  of  gravity. 


CENTRE    OF    GRAVITY    AND    STABILITY.  61 


Examples. 

1.  Required  the  point  of  application  of  the  resultant  of  three  equal 
weights,  applied  at  the  vertices  of  a  plane  triangle. 

SOLUTION. 

Let  ABC  (Fig.  30)  represent  the  triangle.  Tlie  resultant  of  the 
•weights  at  B  and  C  will  be  applied  at  D,  the  middle  of  BC.  The 
weight  acting  at  D  being  double  that  at  A,  the  total  resultant  will  be 
applied  at  G,  making  OA  =  2  OD ;  hence,  the  required  joint  is  the 
centre  of  gravity  of  the  triangle. 

2.  Required  the  point  of  application  of  the  resultant  of  a  system  of 
equal  parallel  forces,  applied  at  the  vertices  of  a  regular  polygon  ? 

Am.  At  the  centre  of  the  polygon. 

3.  Parallel  forces  of  3,  4,  5,  and  6  lbs.,  are  applied  at  the  successive 
vertices  of  a  square,  whose  side  is  12  inches.  At  what  distance  from 
the  first  vertex  is  the  point  of  application  of  their  resultant? 

SOLUTION. 

Take  the  sides  of  the  square  through  the  first  vertex  as  axes ;  call 
the  side  through  the  first  and  second  vertex,  the  axis  of  X,  and  that 
through  the  first  and  fourth,  the  axis  of  Y.     We  shall  have,  from 

Formulas  (21,  22), 

4  X  12  +  0  X12 


=  6 

fi  X  12  -f-  5  X  12 
and 


18 
6X12  +  5X12       22 


^'-  18  3- 

Denoting  the  lequired  distance  by  d,  we  have, 

d  =  Vx;'  +  y;'  =  9.475  in.     Ans. 
4.  Seven  equal  forces  are  applied  at  seven  of  the  vertices  of  a  cube. 
What  is  the  distance  of  the  point  of  application  of  their  resultant 
from  the  eighth  vertex  ? 

SOLUTION, 

Take  the  eighth  vertex  as  the  orgin  of  co-ordinates,  and  the  three 
edges  passing  through  it  as  axes.  We  shall  have,  from  Equations 
(21,  22,  23),  denoting  one  edge  of  the  cube  by  a, 

.r,  =1*7,  ?/,  =  *«,  and  r,  =  ^a. 
Denoting  the  required  distance  by  d,  we  have, 


d  =  Vijy  +  y;'  +  z,^  =  ^a  V'6.     Ans. 
5.  Two  isosceles  triangles  are  constructed  on  opposite  sides  of  the 


62  MECHANICS. 

base  Z>,  having  altitudes  equal  to  li  and  7i',  h  being  greater  than  K . 
Where  is  the  centre  of  gravity  of  the  space  within  the  two  triangles  ? 

SOLUTION. 

It  must  lie  on  the  altitude  of  the  greater  triangle.  Take  the  com- 
mon base  as  an  axis  of  moments ;  then  will  the  moments  of  the  tri- 
angles be  \hh  X  \li,  and  \h1i  X  ^ti  ;  and  from  Fonnula  (21),  we  have, 

That  is,  the  centre  of  gravity  is  on  the  altitude  of  the  greater  tri- 
angle, at  a  distance  from  the  base  equal  to  one-third  of  the  difference 
of  the  two  altitudes. 

6.  Where  is  the  centre  of  gravity  of  the  space  between  two  circles 
tangent  to  each  other  internally  ? 

SOLUTION. 

Take  their  common  tangent  as  an  axis  of  moments.  The  centre 
of  gravity  will  lie  on  the  common  normal,  and  its  distance  from  the 
point  of  contact  is  given  by  the  equation, 

Trr'  —  Ttr'^  ~        r-\-  r*      ' 

7.  Let  there  be  a  square,  divided  by  its  diagonals  into  four  equal 
parts,  one  of  which  is  removed.  Required  the  distance  of  the  centre 
of  gravity  of  the  remaining  figure  from  the  opposite  side  of  the 
square.     An%.  fg  of  the  side  of  the  square. 

8.  To  construct  a  triangle,  having  given  its  base  and  centre  of 
gravity. 

SOLUTION. 

Draw  through  the  middle  of  the  base,  and  the  centre  of  gravity,  a 
straight  line ;  lay  off  beyond  the  centre  of  gravity  a  distance  equal  to 
twice  the  distance  from  the  middle  of  the  base  to  the  centre  of  grav- 
ity.   The  point  thus  found  is  the  vertex. 

9.  Three  men  carry  a  cylindrical  bar,  one  taking  hold  of  one  end, 
and  the  others  at  a  common  point.  Required  the  position  of  this 
point,  in  order  that  the  three  may  sustain  equal  portions  of  the 
weight. 

Am.  At  three-fourths  the  length  of  the  cylinder  from  the  first. 


.1 


m 


CENTRE   OF   GRAVITY    AND   STABILITY.  63 

STABILITY   AND    EQUILIBRIUM. 
Stable,  Unstable,  and  Indifierent  Equilibrium. 

56.  A  body  is  in  stable  equilibrium  Avhen,  on  being 
slightly  disturbed  from  a  state  of  rest,  it  has  a  tendency  to 

return  to  that  state.    This  will  be  the  case  when    0<j^ -IP 

the  centre  of  gravity  of  the  body  is  at  its  lowest 

point.     Let  ^  be  a  body  suspended   from  an 

axis  0,  about  which  it  is  free  to  turn.     When 

the  centre  of  gravity  of  A  lies  vertically  below 

the  axis,  it  is  in  equilibrium,  for  the  weight  of        Fig.  38. 

the  body  is  exactly  counterbalanced  by  the  resistance  of 

the  axis.     Moreover,  the  equilibrium  is  stable ;  for  if  the 

body  be  deflected  to  A',  its  weight  acts  with  the  lever  arm 

OF  to  restore  it  to  its  position  of  rest,  A . 

A  body  is  in  mistable  equilibrium  when  on  being  slightly 
disturbed  from  its  state  of  rest,  it  tends  to  depart  still 
farther  from  it.  This  will  be  the  case  when  the  centre  of 
gravity  of  the  body  occupies  its  highest  position. 

Let  ^  be  a  sphere,  connected  by  an  inflexible  rod  with 
the  axis  0.  When  the  centre  of  gravity  of  A  is  vertically 
above  0,  it  is  in  unstable  equilibrium ;  for,  if 


the  sphere  be  deflected  to  the  position  A',  its     M^     \a^ 


I 


weight   will   act   with   the  lever   arm    OP    to 
increase  the  deflection.     The  motion  continues 
till,  after  a   few   vibrations,   it    comes  to   rest      w        r 
below  the  axis.     In  this  last  position,  it  is  in       Fig.  39. 
stable  equilibrium. 

A  body  is  in  iiidiffereiit,  or  neutral,  equilibrium  when  it 
remains  at  rest,  wherever  it  may  be  placed.  This  is  the 
case  when  the  centre  of  gravity  continues  in  the  same  hori- 
zontal plane  on  being  slightly  disturbed. 

Let  A  be  a  sphere,  supported  by  a  liorizontal  axis   OP 


C4 


MECHANICS. 


through  its  centre  of  gravity.     Then,  in  whatever  position 
it  may  be  placed,  it  will  have  no  tendency 
to  change  this  position;  it  is,  therefore,  q 
in  indifferent,  or  neutral  equilibrium. 

In  the  figure,  A,  B,  and  C,  represent  a 
cone  in  positions  of  stable,  unstable,  and  ^^s-  ^o. 

indifferent  equilibrium. 


Fig.  41. 

If  a  Avheel  be  mounted  on  a  horizontal  axis,  about  which 
it  is  free  to  turn,  the  centre  of  gravity  not  lying  on  the 
axis,  it  will  be  in  stable  equilibrium,  when  the  centre  of 
gravity  is  directly  below  the  axis ;  and  in  unstable  equi- 
librium when  it  is  directly  above  the  axis.  When  the  axis 
passes  through  the  centre  of  gravity,  it  will,  in  every  posi- 
tion, be  in  neutral  equilibrium. 

We  infer,  from  the  preceding  discussion,  that  when  a 
body  at  rest  is  so  situated  that  it  cannot  be  disturbed  from 
its  position  without  raising  its  centre  of  gravity,  it  is  in  a 
state  of  stable  equilibrium;  when  a  slight  disturbance 
depresses  the  centre  of  gravity,  it  is  in  a  state  of  unstable 
equilibritini ;  when  the  centre  of  gravity  remains  con- 
stantly in  the  same  horizontal  plane,  it  is  in  a  state  of 
neutral  equilibrium. 

This  principle  holds  tnie  in  the  combinations  of  wheels 
and  other  pieces  used  in  machinery,  and  indicates  the  im- 
portance of  balancing  these  elements,  so  that  their  centres 
of  gravity  may  remain  in  the  same  horizontal  planes. 


CENTRE    OF   GKAVITY   AND    STABILITY.  65 

Stability  of  Bodies  on  a  Horizontal  Plane. 

57.  A  body  resting  on  a  horizontal  piano  may  touch  it  in 
one,  or  in  more  than   one  point.     In  the  latter  case,  the 
salient  polygon,  formed   by  joining 
the   extreme   points   of    contact,  as                /i\\ 
abed,  is  called  xhe  polygon  of  support.  /  i  \\        

When  the  direction  of  the  weight        /    /AT"\'/      / 

of   the   body,  that  is,   the   vertical    /     ^ ^     / 

through  its  centre  of  gravity,  pierces  Fig.  42. 

the  plane  within  the  polygon  of  support,  the  body  is  stable, 
and  will  remain  in  equilibrium,  unless  acted  upon  by  some 
other  force  than  the  weight  of  the  bod3\  In  this  case,  the 
body  will  be  most  easily  overturned  about  that  side  of  the 
polygon  of  support  which  is  nearest  to  the  line  of  direction 
of  the  weight.  The  moment  of  the  weight,  with  respect  to 
this  side,  is  called  the  moment  of  stability.  Denoting  the 
weight  of  the  body  by  W,  the  distance  from  its  line  of  direc- 
tion to  the  nearest  side  of  the  polygon  of  support  by  ?% 
and  the  moment  of  stability  by  S,  we  have, 

S=  Wr. 

The  moment  of  stability  is  the  moment  of  the  least 
extraneous  force  that  is  capable  of  overturning  the  body. 
The  weight  of  a  body  remaining  the  same,  its  stability 
increases  with  r.  If  the  polygon  of  support  is  a  regular 
polygon,  the  stability  will  be  greatest,  other  things  being 
equal,  when  the  direction  of  the  weight  passes  through  its 
centre.  The  area  of  the  polygon  of  support  remaining 
constant,  the  stability  will  be  greater  as  the  polygon 
approaches  a  circle.  The  polygon  of  support  being  regu- 
lar, but  variable  in  area,  tlie  stability  will  increase  as  the 
area  increases  :  low  bodies  with  extended  bases,  are  more 
stable  than  high  bodies  with  narrow  bases. 


66  MECHANICS. 

When  ilie  direction  of  the  weight  passes  without  the 
polygon  of  support,  the  body  is  unstable,  and  unless  sup- 
ported by  some  other  force  than  the  weight,  it  will  turn 
about  the  side  nearest  the  direction  of  the  weight.  In  this 
case,  the  product  of  the  weight  into  the  distance  from  its 
direction  to  the  nearest  side  of  the  polygon,  is  called  the 
moment  of  instability.      Denoting  this  moment  by  /,  we 

have,  as  before, 

/=  Wr, 

The  moment  of  instability  is  equal  to  the  least  moment 
of  a  force  that  can  prevent  the  body  from  overturning. 

If  the  direction  of  the  weight  intersect  any  side  of  the 
polygon  of  support,  the  body  will  be  in  a  state  of  equilib- 
riuin  bordering  on  rotation  about  that  side. 

If  the  resultant  of  all  the  forces  acting  on  a  body, 
including  its  weight,  be  oblique  to  the  supporting  plane,  it 
may  be  resolved  into  two  components,  one  perpendicular 
to  the  plane  and  the  other  parallel  to  it.  The  former  is 
counteracted  by  the  reaction  of  the  plane ;  the  latter  tends 
to  make  the  body  to  slide  along  the  plane.  Hence  the  im- 
portance of  making  the  resultant  as  nearly  normal  to  the 
supporting  plane  as  possible. 

These  principles  find  application  in  the  arts,  more  espe- 
cially in  Engineering  and  Architecture.  In  structures 
intended  to  be  stable,  the  foundation  should  be  as  broad  as 
is  consistent  with  the  general  design  of  the  work,  that  the 
polygon  of  support  may  be  as  large  as  possible.  The 
pieces  for  transmitting  pressures  should  be  so  combined 
that  the  pressures  may  be  as  nearly  normal  to  the  bearing 
surfaces  as  possible,  and  their  lines  of  direction  should 
pass  as  near  the  centres  of  the  polygons  of  support  as  may 
be.  Hence,  joints  should  be  made  as  nearly  normal  to  the 
pressures  as  possible. 


CEin'RE   OF   GRAVITY   AND   STABILITY.  67 

In  the  construction  of  machinery  the  centres  of  gravity 
of  rotating  pieces  should  be  in  their  axes,  otherwise  there 
will  result  an  irregularity  of  motion,  which,  besides  making 
the  machine  work  imperfectly,  will  ultimately  destroy  the 
machine  itself. 

In  loading  cars,  wagons,  &c.,  we  should  throw  the  centre 
of  gravity  of  the  load  as  near  the  track  as  possible.  This 
is  partially  effected  by  placing  the  heavier  articles  at  the 
bottom  of  the  load. 


Pressure  ot  one  body  on  another. 

58.  Let  A  be  a  movable  body  pressed  against  a  fixed 
body  B,  and  touching  it  at  a  single  point.  In  order  that 
A  may  be  in  equilibrium,  the  result- 
ant of  all  the  forces  acting  on  it,  in- 
cluding its  weight,  must  pass  through 
the  point  of  contact,  P' ;  otherwise 
there  would  be  a  tendency  to  rotation 
about  P',  which  would  be  measured 
by  the  moment  of  the  resultant  with 
respect  to  this  point.     Furthermore,  Fig.  43. 

the  direction  of  the  resultant  must  be  normal  to  the  sur- 
face of  B  at  the  point  P',  else  the  body  A  would  have  a 
tendency  to  slide  along  the  body  B,  which  tendency  would 
be  measured  by  the  tangential  component.  The  pressure 
on  B  develops  a  force  of  reaction,  Vv-hich  is  equal  and 
directly  opposed  to  it.  The  resultant  of  all  the  forces  in- 
cluding the  reaction  must  be  equal  to  zero  (Art.  41).  That 
is,  tvhen  a  body,  resting  un  anotlicr  and  acted  upon  hy  any 
numhcr  of  forces,  is  in  equilibrium,  the  resultant  of  all  the 
forces  called  i?ito  play  is  equal  to  0. 

If  all  the  forces  called  into  play  are  taken  into  account, 


68 


MECHANICS. 


tlie  algebraic  sums  of  tlieir  7noments  tvith  respect  to  any 
three  rectangular  axes  loill  he  separately  equal  to  0. 

If  the  bodies  A  and  B  touch  in  more  than  one  point, 
the  polygonal  figure  formed  by  uniting  the  extreme  points 
of  contact  may  be  called  the  polygon  of  contact.  In  this 
case,  the  resultant  of  all  the  forces  must  pass  Avithin  the 
polygon  of  contact. 


Practical  Problems. 

A  horizontal  beam  AB,  which  sustains  a  load,  is  supported  on 


a  pivot  at  A,  and  by  a  cord  BE,  the  point  E  being  vertically  over  A. 
Required  the  tension  of  BE,  and  the  ver-      ^ 
tical  pressure  on  .4. 


SOLUTION. 

Denote  the  weight  of  the  beam  and 
load  by  W,  and  suppose  its  point  of  ap- 
plication to  be  G.  Denote  GA  by  p,  the 
perpendicul.ir  distance,  AF,  from  A  to 
BE,  by  ;/,  and  the  tension  of  the  cord 
by  t.  If  we  take  J.  as  a  centre  of  mo- 
ments, we  have,  in  case  of  equilibrium, 


=.B 


Wp  =  tp'  ; 


wl 


Or,  denoting  the  angle  EBA  by  a,  and  the  distance  AD  by  5,  we 
have, 


2^'  —b^ma. 


t=  Wr 


V 


b  sin  a' 

To  find  the  vertical  pressure  on  A,  resolve  t  into  components, 
parallel  and  perpendicular  to  AB.  We  have  for  the  latter  compo- 
nent, denoted  by  t\ 

b 
The  vertical  pressure  on  A,  plus  the  weight  W,  must  be  equal  to  t. 
Denoting  the  vertical  pressure  by  J*,  we  liavo, 

P  ^      -n.  T,r/  P  .\  TT^/P   —  b^ 


P+  w=  w 


or,P=Tr(-f 
DG 


)-^{^) 


P=W 


AD' 


CENTRE    OF   GRAVITY    AND    STABILITY. 


69 


O 

Fig.  45. 


When  DC=0;  or,  when  D  and  C  coincide,  the  vertical  pressure 
isO. 

2.  A  rope,  AD,  supports  a  pole,  DO,  one  end 
of  which  rests  on  a  horizontal  plane,  and  from 
the  otlier  is  suspended  a  weight  W.  Required 
the  tension  of  the  rope,  and  the  thrust,  or  pres- 
sure, on  the  pole,  its  weight  being  neglected. 

SOLUTION. 

Denote  the  tension  of  the  rope  by  t,  the  pressure  on  the  pole  by  p, 
the  angle  ADO  by  a,  and  the  angle  OD  Why  ft.  • 

There  are  three  forces  acting  at  D,  which  hold  each  other  in  equi- 
librium ;  the  weight  W,  acting  downward,  the  tension  of  the  rope, 
acting  from  D,  toward  A,  and  the  reaction  of  the  pole,  acting  from 
0  toward  D.  Lay  off  Dd,  to  represent  the  weight,  and  complete 
the  parallelogram  doaD ;  then  will  i)a  represent  the  tension  of  the 
rope,  and  Do  the  thrust  on  the  pole. 

From  Art.  33,  we  have, 

•     /?     •                    .    /       w  si^  /^ 
sm  /3  :  sm  a ;  . .  t  =■  W  — . 


t:  W 

We  have,  also,  from  the  same  principle, 
j9  :  TT :  :  sin  (<x  -f  /5)  :  sin  <a: ; 


P 


^sin(a+_^ 


If  the  rope  is  horizontal,  we  have  a  =  90°  —  /?,  which  gives, 

W 


t  =  TFtany^,  and  p  = 


cos/i ' 


3.  A  beam  FB,  is  suspended  by  ropes  attached  at  its  extremities, 
and  fastened  to  pins  A  and  H.    Required  the  tensions  of  the  ropes. 

SOLUTION. 

Denote  the  weight  of  the  beam  and  its  load  by  W,  and  let  c  be 
its  point  of  application.  Denote  the  tension  of  the  rope  BH,  by  t, 
and  that  of  FA  by  t'.  The  forces  in  equilib- 
rium, are  W,  t,  and  t'.  The  plane  of  these 
forces  must  be  vertical,  and  further,  the 
directions  of  the  forces  must  intersect  in  a 
point.  Produce  AF,  and  BR,  till  they  inter- 
sect in  K,  and  draw  Kc  ;  take  Kc,  to  repre- 
sent the  weight  of  the  beam  and  its  load, 
and  complete  the  parallelogram  KbCf;  then 


70 


MECHAI^ICS. 


will  Kb  represent  t,  and  Kfyf'iW  represent  t'.    Denote  the  angle  cKB 
by  a,  and  cKF  Xiy  ft.    We  shall  have,  as  in  the  last  problem, 

sin  ft 


And, 


W:  t:  :sin(a:  +  ^)  :  sin/?; 


W :  t'  :  :  sin(cY-f  /?)  :  sin  a 


t=W 


sin(a  -f  ft ' 


r  =  TF- 


sin(«  +  ft) ' 

4.  A  gate  AH,  is  supported  at  0  on  a  pivot,  and  at  A  by  a  hinge, 
attached  to  a  post  AB.  Required  the  pressure  on  the  pivot,  and  the 
tension  of  the  hinge. 

SOLUTION. 

Denote  the  weight  of  the  gate  and  its  load,  by  TF,  and  let  Cbe  its 
point  of  application.  Produce  the  vertical  through  (7,  till  it  intersects 
the  horizontal  through  A  in  D,  and  draw  BO. 
Then  will  AB  and  DO  be  the  directions  of  the  re- 
quired components  of  W.  Lay  off  Be,  to  represent 
TF,  and  complete  the  parallelogram,  Bcoa;  then 
will  Bo  represent  the  pressure  on  jO,  and  aB  the 
tension  on  the  hinge,  A.  Denoting  the  angle  oBc 
by  a,  the  pressure  on  the  pivot  by  p,  and  on  the 
hinge  by  p',  we  have, 

p  ■=.  ^^^^,  and  j9  =  ^  sm  a. 


"B 


Fig.  47. 


cosa 
If  we  denote  OE  by  h,  and  BE  by  h,  w^e  shall  have, 


Hence, 


V 


and  sin  a 


and  p'  = 


pb 


Vb''  +  ii' 


5.  Having  two  rafters,  AG  and 
BC,  abutting  in  notches  of  a  tie- 
beam  AB,  it  is  required  to  find  the 
pressure,  or  thrust,  on  the  rafters, 
and  the  direction  and  intensity  of 
the  pressure  on  the  joints  at  the 
tie-beam. 


I'r--i 


Fig.  48. 


SOLUTION. 

Denote  the  weight  of  the  rafters  and  tlieir  load  by  2id  ;  we  may 
regard  this  weight  as  made  up  of  three  parts — a  weight  w,  applied  at 


CENTRE   OF   GKAVITY    AND    STABILITY.  71 

C,  and  two  equal  weights  Iw,  applied  at  A  and  B  respectively.  De- 
note the  half  span  AL  by  s,  the  rise  CL  by  A,  and  the  length  of  the 
rafter  ^ICor  CB  by  I.  Denote,  also,  the  angle  CBL  by  a,  the  thrust 
on  each  rafter  by  t,  and  the  resultant  pressure  at  each  of  the  joints 
A  and  B  by  p. 

Lay  off  Co  to  represent  the  weight  w,  and  complete  the  parallelo- 
gram Cboa  ;  then  will  Ca  and  Cb  represent  the  thrust  on  the  rafters ; 
and,  since  Cboa  is  a  rhombus,  we  have, 

I  .  w  wl 

is\no:  =  i^w  .-.  t  =  —-, —  =  ?n-- 

2  smo:       2h 

Conceive  t  to  be  applied  at  A,  and  there  resolve  it  into  components 
parallel  to  CL  and  LA  ;  we  have,  for  these  components, 

t  sm  a  =  \w^  and  t  cos  a  =  — . 

The  latter  component  gives  the  strain  on  the  tie-beam,  AB. 
To  find  the  pressure  on  the  joint,  we  have,  acting  downward, 
the  forces  \w  and  ^w^  or  the  single  force  w,  and,  acting  from    L 

toward  A,  the  force  ^ ;  hence. 


If  we  denote  the  angle  DAEhy  /j,  we  have  from  the  right-angled 
triangle  DAE, 

BE      1CS  s 

The  joint  should  be  perpendicular  to  the  force  p,  that  is,  it  should 
make  with  the  horizon  an  angle  whose  tanfjent  is  :^ . 

G.  In  the  last  problem  suppose  the  rafters  to  abut  against  the  wall. 
Required  the  least  thickness  that  must  be  given  to  it  to  prevent  it 
from  being  overturned. 

SOLUTION. 

Denote  the  weight  thrown  on  the  wall  by  w,  the  length  of  wall 
that  sustains  the  pressure  p  by  l',  its  height  by  7i',  its  thickness  by  x, 
and  the  weight  of  each  cubic  foot  of  the  wall  by  w';  then  will  the 
weight  of  this  part  be  w'h'l'x. 

The  force  —r  acts  with  an  arm  of  lever  7i'  to  overturn  the  wall 

2/7. 

about  its  lower  and  outer  edge ;  this  force  is  resisted  by  the  weight 


7:3  MECHANICS. 

w  +  w'h'l'x,  acting  through  the  centre  of  gravity  of  the  wall  with  a 
lever  arm  equal  to  ^i:.  If  there  be  an  equilibrium,  the  moments  of 
these  two  forces  are  equal,  that  is, 

ins       ^,       ,      ,      ,,  ,y,  ,  X         wsJl' 

—  X7i'  =  {wi-w  h'l'x)  — ,  or  -j—  =  iox-i-  w'h'l'x^. 

Reducing,  we  have,  x^  A -— ,  x  =  — ^-^ ; 

will'         wl'h 


I  ws 


".±x/.-^  + 


2w'h'l'  -^  V  w'/W   '   4:W'Vi'H'-' ' 

7.  A  sustaining  wall  has  a  cross  section  in  the  form  of  a  trapezoid, 

the  face  on  which  the  pressure  is  thrown  being  vertical,  and  the 

opposite  ftice  having  a  slope  of  «z^  perpendi-  ^ 

cular  to  one  horizontal.     Required  the  least  a 

lliickness  that  must  be  given  to  the  wall  at  /• 

top,  that  it  may  not  be  overturned  by  a  lior-  /  ,:     ,< 

izontal  pressure,  whose  point  of  application  /  ;  i     i 

is  at  a  distance  from  the  bottom  of  the  wall         /    i  ;     ; 
equal  to  one-third  its  height.  D  E  F  G-   C 

Fig.  49. 

SOLUTION. 

Pass  a  plane  through  the  edge  A  parallel  to  BC,  and  consider  a 
portion  of  the  wall  whose  length  is  one  foot.  Denote  the  pressure 
on  this  by  P,  the  height  of  the  wall  by  C^h,  its  thickness  at  top  by  x, 
and  the  weight  of  a  cubic  foot  by  w.  Let  fall  from  the  centres  of 
gravity  0  and  0'  of  the  two  portions,  j^erpendiculars  OG  and  O'E, 
and  take  the  edge  D  as  an  axis  of  moments.  The  weight  of  the  por- 
tion ABGF  is  equal  to  (Swhx,  and  its  lever  arm,  ])0,  is  equal  to 
/i  +  \x.  The  weight  of  the  portion  ^i>i^ is  ^wh"^,  and  its  lever  arm, 
DE,  is  |7i.  In  case  of  equilibrium,  the  sum  of  the  moments  of  their 
weights  must  be  equal  to  the  moment  of  P,  whose  lever  arm  is  2h. 
Hence 

Qwlixiji  +  \x)  +  ^wK'  xyi=  Px  27i ; 

or,  ewJtx  +  Swx''  +  2w?t''  =  2P. 

2(P  —  wJi') 
Whence,  x""  +  2hx  =  -^— ^ ' ; 


-/^:^-\/-— T^— +^^^ 


'2{P-joh-') 
dw 

8.  Required  the  conditions  of  stability  of  a  square  pillar  acted  on 
by  a  force  oblique  to  the  axis,  and  applied  at  the  centi-e  of  gravity 
of  the  upper  base. 


CENTRE   OF   GRAVITY    AND   STABILITY. 


73 


SOLUTION. 

Denote  the  intensity  of  the  force  by  P,  its  inclination  to  the  verti- 
cal by  a,  the  breadth  of  the  pillar  by  2a,  its  height  by  x,  and  its 
weight  by  W.  Through  the  centre  of  gravity  of 
the  pillar  draw  a  vertical  AC,  and  lay  off  ylC  equal 
to  W;  prolong  PA  and  lay  off  AB  equal  to  P; 
complete  the  parallelogram  ABDC,  and  prolong 
the  diagonal  till  it  intersects  IIG  at  F.  If  F  is  be- 
tween H  and  G,  the  pillar  will  be  stable ;  if  at  H, 
it  will  be  indifferent ;  if  without  H,  it  will  be  un- 
stable. To  find  an  expression  for  FO,  draw  DE 
perpendicular  to  AO.  From  the  similar  triangles 
ADE  and  AFG,  we  have, 


/jc 

'M 

.....JE 


m 


Fig.  50. 


AE  :  AG  :  :  DE  :  FG ;  :.  FG  = 

P&ina,  and  AE 


But  AG  =  X,  DE 
have. 


AGXDE 
AE       • 
W-i-  Pcosa,  hence,  we 


FG  = 


Px  sin« 


W  -f  Pcosa ' 

And,  since  HG  equals  a,  we  have  the  following  conditions  for  sta- 
bility, indifference,  and  instability,  respectively : 

Px  sino: 


a> 


a  < 


W  -f  Pcosa ' 

Px  sma 
TF  4- Pcosa' 

Px  sina 
IT -f  Pcosa* 
4 


CHAPTER  IV. 

ELEMENTARY   MACHINES. 

Definitions  and  General  Principles. 

59.  A  MACHINE  is  a  contrivance  by  means  of  which  a 
force  applied  at  one  point  is  made  to  produce  an  effect  at 
some  other  point. 

The  applied  force  is  called  the  power,  and  the  force  to  be 
overcome  the  resistance;  the  source  of  the  power  is  called 
the  motor. 

Some  of  the  more  common  motors  are  muscular  effort, 
as  exhibited  by  man  and  beast  in  various  kinds  of  work ; 
the  weight  and  living  force  of  water,  as  shown  in  the 
various  kinds  of  water-mills;  the  expansive  force  of  vapors 
and  gases,  as  displayed  in  steam  and  caloric  engines ;  the 
force  of  air  in  motion,  as  exhibited  in  the  windmill,  and 
in  the  propulsion  of  sailing  vessels;  the  force  of  r.iagnetic 
attraction  and  repulsion,  as  shown  in  the  magnetic  tele- 
graph and  various  magnetic  machines ;  the  elastic  force  of 
springs,  as  shown  in  watches  and  various  other  machines. 
Of  these  the  most  important  are  steam  and  water  power. 

Work. 

60.  Work  is  the  effect  produced  by  a  force  in  overcoming 
a  resistance ;  it  implies  the  simultaneous  existence  of  both 
pressure  and  motion. 

The  measure  of  the  work  done  by  a  force,  is  the  product 
of  the  effective  pressure,  by  the  distance  through  which  it 
is  exerted. 


ELEMENTARY    MACHINES.  7o 

Machines  simply  transmit  and  modify  the  action  of 
forces.  They  add  nothing  to  the  work  of  the  motor ;  on 
the  contrary,  they  absorb  and  render  inefficient  much  of 
that  which  is  impressed  on  them.  For  example,  in  a 
Water-mill,  only  a  portion  of  the  work  expended  by  the 
motor  is  transmitted  to  the  machine,  on  account  of  the 
imj^erfect  manner  of  applying  it,  and  of  this  portion  a 
large  part  is  absorbed  and  rendered  practically  useless  by 
resistances,  so  that  only  a  small  portion  of  the  work  ex- 
pended by  the  motor  becomes  effective. 

Of  the  ayiMed  ivorh,  a  part  is  expended  in  overcoming 
friction,  stiffness  of  cords,  lands,  or  chains,  resistance  of 
the  air,  adhesion  of  the  parts,  &c.  This  goes  to  wear  out 
the  machine.  A  second  portion  is  expended  in  overcoming 
shocks,  arising  from  the  nature  of  the  work  to  be  accom- 
plished, as  well  as  from  imperfect  connection  of  the  parts, 
and  from  want  of  hardness  and  elasticity  in  the  connecting 
pieces.  This  also  goes  to  strain  and  wear  out  the  machine, 
and  to  increase  the  waste  already  mentioned.  There  is 
often  a  waste  of  work  arising  from  a  greater  supply  of  mo- 
tive power  than  is  required  to  attain  the  desired  result. 
Thus,  in  the  movement  of  a  train  of  cars,  the  excess  of 
work  of  the  steam,  above  what  is  necessary  to  bring  the 
train  to  the  station,  is  wasted,  being  consumed  by  the 
application  of  brakes,  an  operation  that  not  only  wears  out 
the  brakes,  but  also,  by  creating  shocks,  ultimately  de- 
stroys the  cars  themselves. 

Such  are  some  of  the  sources  of  loss  of  work.  A  part 
of  these  may,  by  judicious  combinations,  be  greatly  dimin- 
ished; but,  under  the  most  favorable  circumstances,  there 
is  a  continued  loss  of  work,  which  requires  a  continued 
supply  of  power. 

In   a  machine,  the  quotient  obtained  bv  dividing  the 


76  MECHAN^ICS. 

quantity  of  useful,  or  effective  work,  by  the  quantity  of 
ajypUcd  v'ork,  is  called  the  modulus  of  the  machine.  As 
the  resistances  are  diminished,  the  modulus  increases,  and 
the  machine  becomes  more  perfect.  Could  the  modulus 
become  equal  to  1,  the  machine  would  be  perfect.  Once 
set  in  motion,  it  would  continue  to  move  forever,  realizing 
the  idea  ot  perpetual  motion.  It  is  needless  to  say  that, 
until  the  laws  of  nature  are  changed,  no  such  realization 
can  be  looked  for. 

Trains  of  Mechanism. 

61.  A  machine  usually  consists  of  an  assemblage  of 
moving  pieces  called  elements,  kept  in  position  by  a  con- 
nected system  called  ^  frame.  Of  the  moving  pieces,  that 
which  receives  the  power  is  called  the  recipient,  that  which 
performs  the  work,  is  called  the  operator  or  tool,  and  the 
connecting  pieces  constitute  what  is  called  a  train  of  me- 
chanism. Of  two  consecutive  elements,  that  which  imparts 
motion  is  called  a  driver,  and  that  which  receives  motion 
is  called  a  folloiver.  Each  piece,  except  the  extremes,  is  a 
follower,  with  respect  to  that  which  precedes,  and  a  driver, 
with  respect  to  that  which  follows. 

In  studying  a  train  of  mechanism  Ave  find  the  relation 
between  the  power  and  resistance  for  each  element  neglect- 
ing hurtful  resistances.  We  then  modify  these  results  so 
as  to  take  account  of  all  these  resistances,  such  as  friction, 
adhesion,  stififhess  of  cords,  &c.  Having  found  the  relation 
between  the  power  and  resistance  for  each  piece,  we  begin  at 
one  extreme  and  combine  them,  recollecting,  that  the  resist- 
ance for  each  driver  is  equal  to  the  poioer  for  its  follower. 

We  might  also  find  the  modulus  of  each  element,  and 
take  the  product  of  these  partial  moduli  as  the  modulus 
of  the  machine. 


ELEMENTARY    MACHINES.  77 

We  shall  first  show  the  relations  between  the  power  and 
resistance  in  the  difierent  elements  on  the  supposition  that 
there  are  no  hurtful  resistances. 

The  Mechanical  Powers. 

62.  The  elements  to  which  all  machines  can  be  reduced, 
are  sometimes  called  mechanical j^owers.  They  are  seven  in 
number — viz.,  the  cord,  the  lever,  the  inclined  plane,  the 
pulley,  tli3  wheel  and  axle,  the  screw,  and  the  wedge.  The 
first  three  are  simple  elements ;  the  pulley,  and  the  wheel 
and  axle  are  combinations  of  the  cord  and  lever;  the 
screw  is  a  combination  of  two  inclined  planes  twisted  round 
an  axle ;  and  the  wedge  is  a  simple  combination  of  two 
inclined  planes. 

The  Cord. 

63.  Let  AB  be  a  cord  solicited  by  two  forces,  P  and  II, 
applied  at  its  extremities,  A  and  B.  In  order  that  the  cord 
may  be  in  equilibrium,  it  is  evi-  ^ 

dent,  in  the  first  place,  that  the  -^         ^ 

forces  must  act  in  the  direction  ^^'   ^' 

of  the  cord,  and  in  such  manner  as  to  stretch  it,  otherwise 
the  cord  would  bend;  and  in  the  second  place,  the  forces 
must  be  equal,  otherwise  the  greater  would  prevail,  and 
motion  would  ensue.  Hence,  if  two  forces  applied  at  the 
extremities  of  a  cord  are  in  equilibrium,  the  forces  are  equal 
and  directly  opposed. 

The  tensio7i  of  a  cord  is  the  force  hy  tvhich  any  tivo  of  its 
adjacent  particles  are  urged  to  separate.  If  a  cord  be  so- 
licited in  opposite  directions  by  equal  forces,  its  tension  is 
measured  by  either  force.  If  the  forces  are  unequal,  the 
tension  is  measured  by  the  less. 

Let  AB  be  a  cord  solicited  by  groups  of  forces  applied 


78  MECHANICS. 

at  its  extremities.     In  order  that  these  forces  may  be  in 
equilibrium,  the  resultants  of  the  groups 
at  A  and  B  must  be  equal  and  directly 


opposed.      Hence,   if    Ave   suppose    the      x  \ 

forces  at  each  point  resolved  into  com-  ^'^s-  ^'-• 

ponents  coinciding  with,  and  at  right  angles  to,  AB,  the 
normal  components  at  each  point  must  he  in  equilibrium, 
and  the  resultants  of  the  remaining  components  at  A  and  B 
must  be  equal  and  directly  opposed. 

Let  A  BCD  be  a  cord,  at  the  points  A,  B,  C,  D,  of  which 
groups  of  forces  are  applied.  If  these  forces  are  in  equi- 
librium through  the  inter- 
vention of  the  cord,  there 
must  necessarily  be  an  equi- 
librium at  each  point,  and 
this  whatever  may  be  the 
lengths  of  ^  ^,  ^  C,  and  CD.  ^'"-  ^^• 

If  we  make  these  infinitely  small,  the  equilibrium  will  still 
subsist.  But  in  that  case  the  points  A,  B,  (7,  and  D,  will 
coincide,  and  all  the  forces  will  be  applied  at  a  single  point. 
Hence,  we  conclude,  that  a  system  of  forces  applied  in 
any  manner  at  points  of  a  cord  tvill  be  in  equilibinum, 
luhen,  if  ajjplied  at  a  siiigle  point  without  change  of  in- 
tensity or  direction,  they  will  maintain  each  other  in  equi- 
librium. 

Hence,  cords  in  machinery  simply  transmit  the  action 
of  forces,  without  modifying  their  effects  in  any  other 
manner. 

The  Lever. 

64.  A  lever  is  an  inflexible  bar,  free  to  turn  about  an 
axis,  called  the  fulcrum. 

Levers  are  divided  into  three  classes,  according  to  the 


ELEMENTARY  MACHINE^. 


79 


Fis.  54. 


2d  Class. 


Fig.  55. 
3d  Class. 


relative  positions  of  the  points  of  application  of  the  power 
and  resistance. 

In  the  first  class,  the  fulcrum  is  between  the   power 
and  resistance.     The  ordinary  balance  1st  class. 

is  an  example  of  this   class  of  levers.  ^ 

The  substance  to  be  ^veighed  is  the  re-     I 
sistance ;  the  counterpoising  weight  is    5 
the  power,  and  the  axis  of  suspension  is 
the  fulcrum. 

In  the  second  class,  the  resistance  is 
between  the  power  and  the  fulcrum. 
The  ordinary  nut-cracker  is  an  example 
of  this  class.  The  nut  is  the  resistance ; 
the  power  is  applied  at  the  ends  of  the 
blades,  and  the  fulcrum  is  at  the  hinge. 

In  the  third  class,  the  power  is  be- 
tween the  fulcrum  and  the  resistance. 
A  pair  of  tongs  furnishes  an  example 
of  this  class.      The   resistance   is  the 
substance  seized  between  the  blades; 
the  power  is  applied  at  the  middle  of 
the  blades;  and  the  fulcrum  is  at  the 
hinge. 

Levers  may  be  curved,  or  straight;   Jj 
and  the  power  and  resistance  may  be  Fig.  56. 

either  parallel  or  oblique  to  each  other.  We  shall  suppose 
the  power  and  resistance  to  be  perpendicular  to  the  ful- 
crum ;  for,  if  not  so  situated,  we  might  conceive  each  to  be 
resolved  into  two  components — one  perpendicular,  and 
the  other  parallel,  to  the  axis.  The  latter  would  bend  the 
lever  laterally,  or  make  it  slide  along  the  axis,  developing 
hurtful  resistance,  while  the  former  alone  would  tend  to 
turn  the  lever  about  the  fulcrum. 


80  MECHANICS. 

The  perpendicular  distances  from  the  fulcrum  to  the 
lines  of  direction  of  the  power  and  resistance,  are  called 
lever  arms  of  these  forces.  In  the  bent  lever  MFN,  the 
perpendicular  distances  FA,  and  FB,  are  the  lever  arms 
of  P  and  R. 

To  determine  the  conditions  of 
equilibrium  of  the  lever,  let  us 
denote  the  power  by  P,  the  resist- 
ance by  R,  and  their  lever  arms 
by  p  and  r.  We  have  the  case  of  a 
body  restrained  by  an  axis,  and  if  Fig.  57. 

we  take  this   as   the  axis  of  mo- 
ments, we   shall   have   for  the   condition  of  equilibrium 
(Art.  41), 

Pp  =  Rr;  or,  P  :  R  ::  r  :  p (30) 

That  is,  the  p)otver  is  to  the  resistmice,  as  the  lever  arm  of 
the  resistance,  to  the  lever  arm  of  the  power. 

This  relation  holds  good  for  every  kind  of  lever. 

The  ratio  of  the  power  to  the  resistance  Avhen  in  equi- 
librium, either  statical  or  dynamical,  is  called  the  leverage, 
or  mechanical  advantage. 

When  the  power  is  less  than  the  resistance,  there  is  said 
to  be  a  gai)i  of  power,  hut  a  loss  of  velocity ;  the  space 
passed  over  by  the  power,  in  performing  any  work,  is  as 
many  times  greater  than  that  passed  over  by  the  resistance, 
as  the  resistance  is  greater  than  the  power.  When  the 
power  is  greater  than  the  resistance,  there  is  said  to  be  a 
loss  of  power,  hut  a  gain  of  velocity.  When  tlie  power  and 
resistance  are  equal,  there  is  neither  gain  nor  loss  of  power, 
but  simply  a  change  of  direction. 

In  levers  of  the  first  class,  there  may  be  either  gain  or 
loss  of  power;  in  those  of  the  second  class,  there  is  always 


ELEMENTARY   MACHINES.  81 

gain  of  i:)ower;  in  those  of  the  third  class,  there  is  always 
loss  of  power.  A  gain  of  power  is  always  attended  with  a 
corresponding  loss  of  velocity,  and  the  reverse. 

If  several  forces  act  on  a  lever  at  different  points,  all 
being  perpendicular  to  the  direction  of  the  fulcrum,  they 
will  be  in  equilibrium,  when  the  algebraic  sum  of  their 
moments,  iinth  resjiect  to  the  fulcriim,  is  equal  to  0. 

Among  the  forces  must  be  included  the  weight  of  the 
lever,  which  is  to  be  regarded  as  vertical  force,  applied  at 
the  centre  of  gravity. 

The  pressure  on  the  fulcrum  is  the  resultant  of  all  the 
forces,  including  the  weight  of  the  lever. 

The  Compound  Lever. 

65.  A  compound  lever  is  a  combination  of  simple  levers 
AB,  BC,  CD,  so  arranged  that  the  resistance  in  one  acts 
as    a    power    in    the    next,  ji*  -^^ 

throughout     the     combina-  ■       —      ^ 

tion.  Thus,  a  power  P  pro- 
duces at  ^  a  resistance  W, 
which,  in  turn,  produces  at 
C  a  resistance  R",  and  so  on. 
Let  us  assume  the  notation 

of  the  figure.  From  the  principle  of  the  simple  lever,  we 
have  the  relations, 

P2)  =  R'r",  R'p'  =  R"r',  R"p"  =  Rr. 

Multiplying  these  equations,  member  by  member,  and 
^riking  out  common  factors,  we  have, 

Ppp'jf  =  Rrr'r" ;  or,  P  :  R  :  :  q^r'r"  :  pp'p".  .  .  .  (31) 

And  similarly  for  any  number  of  levers. 

Hence,  in  the  compound  lever,  the  pmoer  is  to  the  resist- 

4'' 


A            -rJ 

1" 

T 

ic         B 

1     i 

Ficr.  58. 


82  MECHANICS. 

ance  as  the  continued  product  of  the  alternate  arms  of  lever, 
commencing  at  the  resistance,  is  to  the  continued  product  of 
the  aUeimate  arms  of  lever,  coni'mencing  at  the  potver. 

By  suitably  adjusting  the  simple  levers,  any  amount  of 
mechanical  advantage  may  be  obtained. 

The  Elbow-joint  Press. 

66.  Let  CA,  BD,  and  BE  represent  bars,  with  hinge- 
joints  at  B  and  D.     The  bar  CA  has  its  fulcrum  at  C,  and 
the  bar  DE  works   through   a      ^ 
guide  between  D  and  E.    When      7^^^^^^^^::;^.  ,?-^^ 

A   is   depressed,  DE  is   forced    /         ^^^^^fecT     '^ 
against  the  upright  F,  so  as  to     |fij3~&^^^^-^^^^^^^^C 

compress  a  body  placed  between 

E  and  F.      This    machine    is  Fig.  59. 

called  the  elboiu-joint  press,  and  is  used  in  printing,  in 
moulding  bullets,  in  striking  coins  and  medals,  in  punch- 
ing holes,  &c. 

Let  P  denote  the  power  applied  at  A,  perpendicular  to 
AC,  Q  the  resistance  in  the  direction  DB,  and  R  the  com- 
ponent of  Q,  in  the  direction  ED.  Let  C  be  taken  as  an 
axis  of  moments,  and  then,  because  P  and  §  are  in  equi- 
librium, we  have, 

P  X  AC=  Q  XF'C,  or,Q=Px~. 

But,  we  have, 

E=  Qcos  BDH. 

Substituting  and  reducing,  we  have, 

R      AC  cos  BDH 


(32) 


P  F'C 

When  B  is  depressed,  cos  BDH  approaches  1,  and  F'C 
continually  diminishes,  that  is,  the  mechanical  advantage 


ELEMENTARY    MACHINES. 


83 


increases ;  and  finally,  when  B  reaches  ER,  it  becomes  infi- 
nite. There  is  no  limit  to  the  amount  of  compression  that 
can  be  obtained,  except  that  fixed  by  the  strength  of  the 
material.  It  is  to  be  observed  that  the  space  through 
which  the  pressure  is  exerted  varies  inversely  as  the  me- 
chanical advantage. 

Weighing   Machines. 

07.  Nearly  all  Weighing  machines  depend  on  the  princi- 
ple of  the  lever;  the  resistance  is  the  weight  to  be  deter- 
mined, and  the  power  is  a  counterpoising  weight  of  known 
value. 

There  are  two  principal  classes  of  weighing  machines : 
in  i\\Q  first  the  lever  arm  of  the  power  is  constant,  and  the 
power  varies ;  in  the  second,  the  power  is  constant,  and  its 
lever  arm  varies.  The  ordinary  balance  is  an  example  of 
the  first  class,  and  the  steelyard  of  the  second. 

The  Common  Balance. 

68.  The  common  balance  consists  of  a  lever,  AB,  called 
the  beam,  having  a  knife-edge  fulcrum,  F,  and  two  scale- 
pans,  D  and  E,  suspended  from  its 
extremities  by  means  of  knife-edge 
joints  at  A  and  B.  The  beam  is 
supported  by  a  standard,  FK,  rest- 
ing on  a  foot-plate,  L.  The  standard 
is  made  vertical  by  levelling  screws 
passing  through  the  foot-plate.  The 
knife-edges  and  their  supports  are 
of  hardened  steel;  and  to  prevent  unnecessary  wear,  an  ar- 
rangement is  made  for  throwing  them  from  their  bearings 
when  not  in  use.  A  needle,  N,  playing  in  front  of  a  grad- 
uated scale,  GH,  shows  the  amount  of  deflection  of  the 
beam. 


Fig.  60. 


84  MECHANICS. 

In  the  finer  balances  employed  in  scientific  investigation, 
many  additional  contrivances  are  introduced,  to  render  the 
machine  more  perfect.  For  a  complete  description  of 
these  balances  the  reader  is  referred  to  more  extended 
treatises. 

A  good  balance  should  fulfil  the  following  conditions : 
1",  it  should  be  true;  2",  it  should  be  stable — that  is,  when 
the  beam  is  deflected  it  should  tend  to  return  to  a  horizon- 
tal position ;  3°,  it  should  be  sensitive — that  is,  it  should  be 
deflected  from  the  horizontal  by  a  small  force. 

In  order  that  a  balance  may  be  true,  its  lever  arms  must 
be  equal  in  length,  and  both  the  beam  and  scale-pans  must 
be  symmetrical  with  respect  to  two  planes  through  the  cen- 
tre of  gravity  of  the  beam,  the  first  plane  being  perpen- 
dicular to  the  beam,  and  the  second  perpendicular  to  the 
fulcrum. 

In  order  that  it  may  be  stable,  the  centre  of  gravity  of 
the  beam  must  be  below  the  fulcrum,  and  the  line  joining 
the  points  of  suspension  of  the  scale-pans  must  not  pass 
above  the  fulcrum. 

In  order  that  it  may  be  sensitive,  the  line  joining  the 
points  of  suspension  must  not  pass  below  the  fulcrum,  the 
lever  arms  must  be  as  long,  and  the  beam  as  light  as 
is  consistent  with  strength  and  stiffness,  the  knife-edges 
must  be  horizontal  and  parallel  to  each  other,  and  the  fric- 
tion at  the  joints  must  be  as  small  as  possible.  The  sensi- 
tiveness of  a  balance  diminishes  as  the  load  increases. 

The  true  weight  of  a  body  may  be  found  by  a  balance 
whose  lever  arms  are  not  equal,  by  means  of  the  principle 
demonstrated  below. 

Denote  the  length  of  the  lever  arms,  by  r  and  r',  and 
the  weight  of  the  body,  by  IF.  When  the  weight  W  is 
applied  at  the  extremity  of  the  arm  r,  denote  the  counter- 


ELEMENTARY    MACHINES.  85 

poising  weight  by  W ;  and  when  it  is  at  the  extremity  of 
the  arm  r',  denote  the  counterpoising  weight  by  W".  We 
shall  have,  from  the  principle  of  the  lever, 

Wr  =  WY,  and  Wr'  =  W"r. 

Multiplying  these  equations,  member  by  member,  we 
have, 

W^rr'  =  W'W'rr';     .:     W=  VW  W"; 
that  is,  the  true  iceiglit  is  equal  to  the  square  root  of  the 
'product  of  the  apparent  iveights. 

A  still  better  method,  and  one  that  is  more  free  from  the 
effect  of  errors  in  construction,  is  to  place  the  body  to  be 
weighed  in  one.  scale,  and  put  weights  in  the  other,  till  the 
beam  is  horizontal ;  then  remove  the  body  to  be  weighed, 
and  replace  it  by  known  weights,  till  the  beam  is  again 
horizontal;  the  sum  of  the  replacing  weights  will  be  the 
weight  required.  If,  in  changing  the  load,  the  positions 
of  the  knife-edges  be  not  changed,  this  method  is  almost 
perfect;  but  this  is  a  condition  difficult  to  fulfil. 

The  Steelyard. 

69.  The  steelyard  is  an  instrument  for  weighing  bodies. 
It  consists  of  a  lever,  AB,  called  the  beam  ;  a  fulcrum,  F; 
a  scale-pan,  D,  attached  at 
the  extremity  of  one   arm ; 

and    a    known    weight,  E,     AR^7"[gJ]i.iiniMHii ';^:-'^i^^-^-^:7"^ 

movable  along  the  other  arm. 

We  shall  suppose  the  weight 

of  ^  to  be  1  lb.    This  instru-       ^  Fig.  gi. 

ment  is  sometimes  more  convenient  than  the  balance,  but 

it  is  not  so  accurate.     Tlie   conditions  of  sensibility  are 

essentially  the  same  as  for  the  balance.     To  graduate  the 

instrument,  place  a  pound-weight  in  the  pan,  D,  and  move 


86 


MECHANICS. 


the  counterpoise  B  till  the  beam  rests  horizontal — let  that 
point  be  marked  1 ;  next  place  a  10  lb.  weight  in  the  pan, 
and  move  the  counterpoise  B  till  the  beam  is  again  hori- 
zontal, and  let  that  point  be  marked  10 ;  divide  the  inter- 
mediate space  into  nine  equal  parts,  and  mark  the  points 
of  division  as  shown  in  the  figure.  These  spaces  may  be 
subdivided  at  pleasure,  and  the  scale  extended  to  any 
desirable  limits.  We  have  supposed  the  centre  of  gravity 
to  coincide  Avith  the  fulcrum ;  when  this  is  not  the  case, 
the  weight  of  the  instrument  must  be  taken  into  account 
as  a  force  applied  at  its  centre  of  gravity.  We  may  then 
graduate  the  beam  by  experiment,  or  we  may  compute  the 
lever  arms,  corresponding  to  different  weights,  by  the  prin- 
ciple of  moments. 

To  weigh  a  body  with  the  steelyard,  place  it  in  the  scale- 
pan,  and  move  the  counterpoise  B  along  the  beam  till  an 
equilibrium  is  established;  the  mark  on  the  beam  will 
indicate  the  weight. 


The  bent  Lever  Balance. 

70.  This  balance  consists  of  a  bent  lever,  ACB ;  a  ful- 
crum, (7;  a  scale-pan,  D  ;  and  a  graduated  arc,  EF,  whose 
centre  is  the  centre  of  motion,  C. 
When  a  weight  is  placed  in  the 
scale-pan,  the  pan  is  depressed,  the 
weight  B  is  raised,  and  its  lever 
arm  increased.  When  the  moments 
of  the  two  forces  become  equal,  the 
instrument  comes  to  rest,  and  the 
weight  is  indicated  by  a  needle 
projecting  from  B,  and  playing  in  front  of  the  arc  FE. 
The  zero  of  the  arc  BF  is  at  the  point  indicated  by  the 
needle  when  there  is  no  load  in  the  pan  D. 


Fig.  62. 


eleme:n^tary  machines. 


87 


The  instrument  may  be  graduated  experimentally  by 
placing  weights  of  1,  2,  3,  &c.,  pounds  in  the  pan,  and 
marking  the  points  at  which  the  needle  comes  to  rest ;  or 
it  may  be  graduated  by  the  principle  of  moments. 

To  weigh  a  body  with  the  bent  lever  balance,  place  it  in 
the  scale-pan,  and  note  the  point  at  which  the  needle 
comes  to  rest ;  the  reading  will  give  the  weight  sought. 


Compound  Balances. 

71.  Compound  balances  are  used  in  weighing  heavy 
articles,  as  merchandise,  coal,  freight  for  shipping,  &c.  A 
great  variety  of  combinations  have  been  employed,  one  of 
which  is  shown  in  the  figure. 

AB  is  a  platform  on  which  the  object  to  be  weighed  is 
placed;  BC  is  a  guard  firmly  attached  to  the  platform; 
the  platform  is  supported 
on  the  knife-edge  ful- 
crum E,  and  the  piece  D, 
through  the  medium  of  a 
brace  CD ;  GF  is  a  lever 
turning  about  the  fulcrum 
F,  and  suspended  by  a  rod 
from  the  point  L  ;  LN  is  a  lever  having  its  fulcrum  at  M, 
and  sustaining  the  piece  Z>  by  a  rod  KH ;  0  is  a  scale-pan 
suspended  from  the  end  N  of  the  lever  LN.  The  instru- 
ment is  so  constructed,  that 

EF  '.GF  w  KM  :  LM ; 
and  KM  is  generally  made  equal  to  ^V  of  MN.     The  parts 
are  so  arranged  that  the  beam  LN  shall  rest  horizontally 
when  no  weight  is  placed  on  the  platform. 

If,  now,  a  body  Q  be  placed  on  the  platform,  a  part  of  its 
weight  will  be  thrown  on  the  piece  Z>,  and,  acting  down- 


88  MECHANICS. 

ward,  will  produce  an  equal  pressure  at  K.  The  remain- 
ing part  will  be  thrown  on  E,  and,  acting  on  the  lever 
FG^  will  produce  a  downward  pressure  at  G,  which  will 
be  transmitted  to  L  ;  but,  on  account  of  the  relation  given 
by  the  above  proportion,  the  effect  of  this  pressure  on  the 
lever  XiV^will  be  the  same  as  though  the  pressure  thrown 
on  E  had  been  applied  directly  at  K.  The  final  effect  is, 
therefore,  the  same  as  though  the  weight  of  Q  had  been 
applied  at  K,  and,  to  counterbalance  it,  a  weight  equal  to 
tV  of  Q  must  be  placed  in  the  scale-pan  0. 

To  w^eigh  a  body,  place  it  on  the  platform,  and  add 
weights  to  the  scale-pan  till  LN  is  horizontal,  then  10 
times  the  sum  of  the  weights  added  will  be  the  weight 
required.  By  applying  the  principle  of  the  steelyard  to 
this  balance,  objects  may  be  weighed  by  using  a  constant 
counterpoise. 

Examples. 

1.  In  a  lever  of  the  first  class,  the  lever  arm  of  the  resistance  is  2| 
inches,  that  of  the  power,  33^,  and  the  resistance  100  lbs.  What 
power  is  necessary  to  hold  the  resistance  in  equilibrium  ?    Ans.  8  lbs. 

3.  Four  weights  of  1,  3,  5,  and  7  lbs.,  are  suspended  from  points  of 
a  straight  lever,  eight  inches  apart.  How  f\ir  from  the  point  of  ap- 
plication of  the  first  weight  must  the  fulcrum  be  situated,  that  the 
weights  may  be  in  equilibrium  ? 

SOLUTION. 

Let  X  denote  the  required  distance.     Then,  from  Art.  (34) 
1  X  a- -f  3(a;  -  8) -f- 5(a;  -  16) -f- 7(.r— 24)  =  0; 
.-.  a;  =  17  in.    Ans. 

3.  A  lever,  of  uniform  thickness,  and  18  feet  long,  is  kept  horizon- 
tal by  a  weight  of  100  lbs.  applied  at  one  extremity,  and  a  force  P 
applied  at  the  other  extremit)^,  so  as  to  make  an  angle  of  30°  with 
the  horizon.  The  fulcrum  is  20  inches  from  the  point  of  application 
of  the  weight,  and  the  weight  of  the  lever  is  10  lbs.  What  is  the 
value  of  P,  and  what  is  the  pressure  on  the  fulcnim  ? 


ELEMEKTARY  MACHIN^ES.  89 

SOLUTION. 

The  lever  arm  of  Pis  equal  to  124  in.  X  sin  30°  =  62  in.,  and  the 
lover  arm  of  the  weight  of  the  lever  is  52  in.     Hence, 

20  X  100  =  10  X  52  -h  P  X  62 ;  :.  P  =  24  lbs.  nearly. 

We  have,  also, 

B  =  VX''  +  Y'  =  V{1 10  +  24  sin  80")--'  +  (24  cos  'SOy. 
.'.  R  =  123.8  lbs. ; 

,  X      20.785        __„ 

and,  cos  a  =  —  =  ,  .  ,   -  =  .16789 ; 

.-.  «  =  80°  20'  02". 

4.  A  heavy  lever  rests  on  a  fulcrum  2  feet  from  one  end,  8  feet  from 
the  other,  and  is  kept  horizontal  by  a  weight  of  100  lbs.,  applied  at 
the  first  end,  and  a  weight  of  18  lbs.,  applied  at  the  other  end.  What 
is  the  weight  of  the  lever,  supposed  of  uniform  thickness  throughout  ? 

SOLUTION. 

Denote  the  required  weight  by  x  ;  its  arai  of  lever  is  3  feet.  We 
have,  from  the  principle  of  the  lever, 

100  X  2  =.x'  X  3  +  18  X  8 ;  .-.  a;  =  18|  lbs.    Anfi. 

5.  Two  weights  keep  a  horizontal  lever  at  rest ;  the  pressure  on 
the  fulcrum  is  10  lbs.,  the  difference  of  the  weights  is  4  lbs.,  and  the 
difference  of  lever  arms  is  9  inches.  What  are  the  weights,  and  their 
lever  arms  ? 

Ans.  The  weights  are  7  lbs.  and  3  lbs. ;  their  lever  arms  are  15} 
in.,  and  6}  in. 

6.  Tlie  apparent  weight  of  a  body  weighed  in  one  pan  of  a  false 
balance  is  5^  lbs.,  and  in  the  oth-er  i)an  it  is  6,^i  lbs.  What  is  the 
true  weight  ? 

W  =  v^Y  X  if  =  6  lbs.    Ans. 

The  Inclined  Plane. 

72.  An  inclined  plane  is  one  that  is  inclined  to  the 
horizon. 

In  tliis  machine,  tlie  power  may  be  a  force  applied  to  a 
body  eitlier  to  prevent  motion  down  the  plane,  or  to  pro- 
duce motion  up  the  plane,  and  tlic  resistance,  the  Aveight  of 
the  body  acting  vertically  downward.     The  power  may  be 


90 


MECHANICS. 


applied  in  any  direction  whatever ;  but  we  shall  suppose 
it  to  be  in  a  vertical  plane,  perpendicular  to  the  inclined 
plane. 

Let  ^^  be  an  inclined  plane,  0  a  body  on  it,  R  its 
weight,  and  P  the  force  necessary  to  hold  it  in  equilibrium. 
In  order  that  these  two  forces  may  33 

keep  the  body  at  rest,  their  result- 
ant must  be  perpendicular  to  AB 
(Art.  58). 

If  the  direction  of  P  is  given,  its  j^^   '^  b^ 
intensity  may  be  found  as  follows  :  Fig.  64. 

draw  OR  to  represent  the  weight,  and  OQ  perpendicular 
to  AB ;  through  R  draw  RQ  parallel  to  OP,  and  through 
Q  draw  ()P  parallel  to  OR;  then  will  OP  represent  the 
required  intensity,  and  OQ  the  pressure  on  the  plane. 

If  the  intensity  of  Pis  given,  its  direction  may  be  found 
as  follows:  draw  OR  and  0(2  as  before ;  with  P  as  a  cen- 
tre, and  the  given  intensity  as  a  radius,  describe  an  arc 
cutting  OQ  in  Q;  draw  RQ,  and  through  0  draw  OP 
parallel,  and  equal  to  RQ ;  it  will  represent  the  direction 
of  the  force  P. 

If  we  denote  the  angle  between  P  and  R  by  9,  and  the 
inclination  of  the  plane  by  a,  we  have  the  angle  ROQ 
equal  to  a,  since  OQ  i^  perpendicular  to  AB,  and  OR  to 
AC,  and,  consequently,  QOP  =  cp  —  a.  From  Art.  33  we 
have, 

P :  P  : :  sin  a :  sin(9  —  a) .  .  .  (33) 

From  which,  if  either  P  or  9  be 
given,  the  other  can  be  found. 

When  the  power  is  parallel  to  the 
plane,  we  have, 

Fig.  65. 


ELEMENTARY  MACHliq^ES.  91 


(p-a  z=  90°, 
or,  sin((p  —  a)  =     1 ; 

1  .  ,  BC 

also,  •sin  a  =   -— p:. 

AB 

Substituting  these  in  the  preceding  proportion,  and 
reducing,  we  have, 

P:  R  ::  BC  :  AB (34) 

That  is,  the  poioer  is  to  the  resistance,  as  the  height  of  the 
'plane  is  to  its  length.  When  power  is  parallel  to  the  base 
of  the  plane,  we  have,  9  —  a  =  90°  —  a ;  whence, 

AC 
sin((p  —  a)  =  cos  a  =  -— ;  B 

AJd 

,  .  BO  ©^ 

also,      sm  a  =  -j^y  ^^\   ! 

Substituting  in  (33),  and  reducing,    Af^ — ^ " " 

we  have.  Fig.  66. 

P  \  R  w  BC  \  AG  .  .  .  .  (35) 

That  is,  the  potver  is  to  the  resistance,  as  the  height  of  the 
plane  is  to  its  base. 

From  the  last  proportion  we  have, 

P  =  R^^=  i2tana. 
AC 

If  a  increase,  the  value  of  P  will  increase,  and  when  a 
becomes  90°,  P  becomes  infinite ;  that  is,  no  finite  horizon- 
tal force  can  sustain  a  body  against  a  vertical  w^all,  without 
the  aid  of  friction. 

Examples. 

1.  A  power  of  1  lb.,  acting  parallel  to  an  inclined  plane,  supports 
a  weight  of  2  lbs.    What  is  the  inclinafion  of  the  plane  ?    Ans.  30° 


02  MECHANICS. 

2.  The  power,  resistance,  and  normal  pressure,  in  the  case  of  an 
inclined  plane,  are,  respectively,  9,  13,  and  6  lbs.  What  is  the  incli- 
nation  of  the  plane,  and  what  angle  does  the  power  make  with  the 
plane  ? 

SOLUTION.       • 

If  we  denote  the  angle  between  the  power  and  resistance  by  <p, 
and  the  inclination  of  the  plane  by  or,  we  have,  from  (Art.  32), 


6  =  VrS''  +  'r  +  2  X  y  X  13  cos  cp; 
.-.  cp  =  156°  S' 20" . 
Also,  from  (Art.  33),  for  the  inclination  of  the  plane, 

G  :  9  :  :  sin  156°  8'  20''  :  sin  n: ;         .-.  a  =  37°  21'  26". 
Inclination  of  power  to  plane  =  <p  —  90°  —  a  =  28°  46'  54".    Ans. 

3.  A  body  is  supported  on  an  inclined  plane  by  a  force  of  10  lbs., 
acting  parallel  to  the  plane ;  but  it  requires  a  force  of  12  lbs.  to  sup- 
port it  when  the  force  acts  parallel  to  the  base.  What  is  the  weight 
of  the  body,  and  the  inclination  of  the  plane  ? 

Am.  The  weight  is  18.09  lbs.,  and  the  inclination  33°  33'  25". 

The  Pulley. 

73.  A  pulley  is  a  wheel  having  a  groove  around  its  cir- 
cumference to  receive  a  cord ;  the  Avheel  turns  on  an  axis 
at  right  angles  to  its  plane,  and  this  axis  is  supported  by  a 
frame  called  a  block.  The  pulley  is  said  to  be  fixed,  when 
the  block  is  fixed,  and  movable,  when  the  block  is  movable. 
Pulleys  are  used  singly,  or  in  combinations. 

Single  Fixed  Pulley. 

74.  In  this  machine  the  block  is  fixed.  Denote  the 
power  by  P,  the  resistance  by  E,  and  the  radius  of  the  pul- 
ley by  r.  It  is  plain  that  both  the  power  and 
resistance  should  be  at  right  angles  to  the  axis. 
Hence,  if  we  take  the  axis  of  the  pulley  as  an 
axis  of  moments,  we  have,  (Art.  41),  in  case  of 
equilibrium, 

Pr  =  Rr;  or,  P  =  R. 


E LE M E JSf TA It Y    M A(J  1  LINES. 


93 


That  is,  the  poiuer  is  equal  to  the  resistance. 
The  effect  of  this  pulley  is  simply  to  change  the  direction 
of  a  force. 

Single  Movable  Pulley. 

75.  In  this  pulley  the  block  is  movable.  The  resistance 
is  applied  by  means  of  a  hook  attached  to  the  block;  one 
end  of  a  rope,  enveloping  the  lower  part  of  the 
jjulley,  is  attached  at  a  fixed  point,  C\  and  the 
power  is  applied  at  its  other  extremity.  We 
shall  suppose,  in  the  first  place,  that  the  two 
branches  of  the  rope  are  parallel. 

Adopting  the  notation  of  the  preceding  arti- 
cle, and  taking  A  as  a  centre  of  movements, 
we  have,  in  case  of  equilibrium  (Art.  41), 

Fx2r-^Jir;       .-.       P  =^  \R. 

That  is,  when  the  power  and  resistance  are  parallel,  the 
power  is  one-half  the  resistance.  The  tension  of  the  cord 
CA  is  the  same  as  that  of  BP.  It  is,  therefore,  equal  to 
one-half  the  resistance.  If  the  resistance  of  the  point  C 
be  replaced  by  a  force  equal  to  P,  the  equilibrium  will  be 
undisturbed. 

Let  the  two  branches  of  the  enveloping  cord  be  oblique 
to  each  other.  Suppose  the  resist- 
ance C  to  be  replaced  by  a  force 
equal  to  P,  and  denote  the  angle  be- 
tween the  two  branches  of  the  rope 
by  2(p.  If  there  is  an  equilibrium 
between  P,  P,  and  R,  we  must  have 

2  .Pcos(p  =  E. 

Draw  the  chord  AB,  and  denote  Fig.  69. 

its  length  by  c  ;  draw,  also,  the  radius  OB.     Then,  because 


94 


MECHAKICS. 


OR  is  perpendicular  to  AB  and  BP  to  OB,  the  angle  Ahr) 
is  one-half  A  CB,  or  equal  to  (p.     Hence, 

C0S9  =  ^c  -H  r  =  — . 

Substituting  in  the  preceding  equation,  and  reducing,  we 
have, 

Pc=Rr;        .-.     P  :  R  ::  r  \  c  .  .  ,  .  (36) 

That  is,  the  j^oiwr  is  to  the  resistance,  as  tJte  radius  of  the 
pulley,  is  to  the  chord  of  the  arc  enveloped  by  the  rope. 

When  the  chord  is  greater  than  the  radius,  there  is  again 
of  mechanical  advantage  ;  Avhen  less,  there  is  a  loss. 

If  the  chord  is  equal  to  the  diameter,  we  have,  as  before, 

P  =  ^R. 


Combination  of  Movable  Pulleys. 

76.  The  figure  represents  a  combination  of  movable 
pulleys,  in  which  there  are  as  many  cords  ^ 
as  pulleys ;  one  end  of  each  cord  is  attached 
at  a  fixed  point,  the  other  end  being  fast- 
ened to  the  hook  of  the  next  pulley  in 
order,  up  to  the  last  cord,  at  the  second  ex- 
tremity of  which  the  power  is  applied. 

Denote  the  tension  of  the  cord  between 
the  first  and  second  jDulley  by  t,  that  of 
the  cord  between  tlie  second  and  third 
pulley  by  t'.  From  tlie  preceding  Article, 
we  have, 

t  =  \R;   t'  =  it;   P=\t'. 

Multiplying  these  equations  together,  member  by  mem- 
ber, and  reducing,  we  have, 


Fig.  70. 


ELEMENTARY    MACHINES.  95 

Had  there  been  n  pulleys  in  the  combination,  we  should 
have  obtained,  in  a  similar  manner, 

P  =  {\Y.R;    .-.  F  :  R  ::  1  '.  T (37) 

That  is,  tJie  power  is  to  the  resistance,  as  1  is  to  2",  n  de- 
noting the  number  of  pulleys. 

Combinations  of  Pulleys  in  Blocks. 

77.  These  combinations  are  effected  in  various  ways. 
In  most  cases,  but  one  rope  is  employed,  which,  being 
attached  to  a  hook  of  one  block,  passes  round  a  pulley  in 
the  other  block,  then  round  one  in  the  first,  and  so  on, 
from  block  to  block,  till  it  has  passed  round  each  pulley  in 

the  system.   The  power  is  applied  at  the  free  end      ,^ ^^ 

of  the  rope.      Sometimes  the  pulleys  in  each 

block  are  placed  side  by  side,  sometimes  one 

above  another,  as  in  the  figure,  in  which  case 

the  inner  ones  are  made  smaller  than  the  outer 

ones.     The  conditions  of  equilibrium  are  the 

same  in  both  cases.     To  deduce  the  conditions 

of  equilibrium  in  the  case  represented,  denote 

the   power    by  P,   and   the   resistance   by  i?.         i^' 

"When  there  is  an  equilibrium,  the  tension  of 

each  branch  of  the  rope  that  aids  in  supporting  ^ 

the  resistance  must  be  equal  to  P ;  but,  since  U 

the   last   pulley  simply  serves   to   change   the        Fig.  71. 

direction  of  the  force  P,  there  will  be  four  such  branches 

in  the  case  considered ;  hence,  we  shall  have, 

4P  =  R,   or,  P  =  IR. 

Had  there  been  01  pulleys  in  the  combination,  there 
would  have  been  n  supporting  branches,  and  we  ^should 
have  had, 

nP  =  R,   or,  P  :  R  ::  1  :  n (38) 


96  MECHANICS. 

That  is,  the  ])oioer  is  to  the  resistance,  as  1  is  to  the  num- 
her  of  branches  of  the  rope  that  supimrt  the  resistance. 

The  principles  already  considered  are  sufficient  to  deter- 
mine the  relation  between  the  power  and  resistance  in  any 
combination  whatever. 

Examples. 

1.  In  a  system  of  six  movable  pulleys,  of  the  kind  described  in 
Art.  7G,  what  weight  can  be  sustained  by  a  power  of  12  lbs.  ? 

Ans,  768  lbs. 

2.  In  a  combination  of  pulleys  in  two  blocks,  when  there  are  six 
pulleys  in  each  block,  what  weight  can  a  power  of  12  lbs.  sustain  in 
equilibrium  ?  Ans.  144  lbs. 

3.  In  a  combination  of  separate  movable  pulleys,  the  resistance  is 
576  lbs.,  and  the  power  that  keeps  it  in  equilibrium  is  9  lbs.  How 
many  pulleys  in  the  combination  ?  Ans.  6. 

4.  In  a  combination  of  pulleys  in  two  blocks,  with  a  single  rope, 
the  power  is  62  lbs.,  and  the  resistance  496  lbs.  How  many  pulleys 
in  each  block?  Ans.  4. 

5.  In  a  combination  of  two  movable  pulleys,  the  inclinations  of  the 
ropes  at  each  pulley  is  60°.  What  is  the  power  required  to  support 
a  weight  of  27  lbs.  V  Ans.  9  lbs. 

The  Wheel  and  Axle. 

78.  The  wheel  and  axle  consists  of  a  wheel,  A,  mounted 
on  an  axle,  B.     The  power  is  applied 
at  one  extremity  of  a  rope  wrapped 
around  the  wheel,  and  the  resistance 
at  one  extremity  of  a  second  rope, 
wrapped  around  the  axle   in  a  con- 
trary direction.     The  whole   is   sup- 
ported by  pivots  projecting  from  the  pig,  .^af 
ends  of  the  axle.     In  deducing  the 
conditions  of  equilibrium,  we  shall  suppose  the  power  and 
resistance  to  be  at  right  angles  to  the  axis. 


ELEMENTARY    MACHINES.  97 

Denote  the  power  by  P,  the  resistance  by  7?,  the  radius 
of  the  wheel  by  r,  and  the  radius  of  the  axle  by  r' .  We 
shall  have,  in  case  of  equilibrium  (Art.  41), 

Pr  =  Rr',  0Y,P:E::r':r....  (39) 

That  is,  the  poiver  is  to  the  resistance,  as  the 
radius  of  the  axle,  to  the  radius  of  the  wheel. 

By  suitably  varying  the  dimensions  of  the 
wheel  and  axle,  any  amount  of  mechanical 
advantage  may  be  obtained. 

If  we  draw  a  line  from  the  point  of  contact  of  the  first 
rope  with  the  wheel,  to  the  point  of  contact  of  the  second 
rope  with  the  axle,  the  power  and  resistance  being  parallel, 
it  will  cut  the  axis  of  revolution  at  the  point  that  divides 
the  line  through  the  points  of  contact  into  parts,  inversely 
proportional  to  the  power  and  resistance.  Hence,  this  is 
the  point  of  application  of  the  resultant  of  these  forces. 
The  resultant  is  equal  to  the  sum  of  the  forces,  and  by  the 
principle  of  moments,  the  pressure  on  each  pivot  may  be 
computed.  When  the  weight  of  the  machine  is  taken  into 
account,  we  regard  it  as  a  vertical  force  applied  at  the 
centre  of  gravity  of  the  wheel  and  axle.  The  pressures 
on  each  pivot  due  to  this  weight  may  be  computed 
separately,  and  the  results  combined  with  those  already 
found. 

Combinations  of  Wheels  and  Axles. 

79.  If  the  rope  of  the  first  axle  be  passed  around  a 

I  second  wheel,  and  the  rope  of  the  second  axle  around  a 
third  wheel,  and  so  on,  a  combination  will  result,  capable 
of  aflTording  great  mechanical  advantage.  The  figure 
represents  a  combination  of  two  wheels  and  axles.  To 
deduce  the  conditions  of  equilibrium,  denote  the  power  by 


98 


MECHANICS. 


rr 


P,  the  resistance  by  R,  the  radius  of  the  first  wheel  by 
r,  that  or  the  first  axle  by  r',  tliat  of  the  second  Avheel  by 
/',  and  that  of  the  second  axle  by  r'". 
If  we  denote  the  tension  of  the  con- 
necting rope  by  t,  this  may  be  regarded 
as  a  power  applied  to  the  second  wheel. 
From  what  was  demonstrated  for  the 
wheel  and  axle,  we  shall  have, 

Pr  =  tr',    and  tr"  =  Rr'". 

Multiplying  these  equations  member 
by  member,  and  reducing,  we  have, 

Prr"  =  Rr'r"';  or,  P  :  R  :  :  r'r"' 

In  like  manner,  were  there  any  number  of  wheels  and 
axles  in  the  combination,  we  might  deduce  the  relation, 

Prr"r^'''  .  .  .  =  Rr'r"'r^'  .  .  .; 
or,  P  :  R  ::  r'r"'r' :  ri'"r'' (40) 

That  is,  the  power  is  to  the  resistance,  as  the  continued 
prochict  of  the  radii  of  the  axles,  to  the  continued  jrroduct 
of  the  radii  of  the  wheels. 

The  principle  just  explained,  is  applicable  to  machinery 
in  which  motion  is  transmitted  from  wheel  to  wheel  by 
bands,  or  belts.  An  endless  band,  called  the  driving  belt, 
passes  around  one  drum  mounted  on  the  axle  of  the  driving 
wheel,  and  around  another  on  that  of  the  driven  wheel. 


The  Crank  and  Axle,  or  Windlass. 
80.  This  machine  consists  of  an  axle,  AB,  and  a  crank, 
BCD.  The  power  is  applied  to  the  crank-handle,  DC, 
and  the  resistance  to  a  rope  wrapped  around  the  axle. 
The  distance,  BC,  from  the  handle  to  the  axis,  is  the 
crank-arm. 


ELEMENTARY   MACHINES. 


99 


The  relation  between  the  power  and  resistance  is  the  same 
as  in  the  wheel  and  axle,  except  that  we  substitute  the 
crank-arm  for  the  radius  of  -p^^ 
the  wheel. 

Hence,  the  poioer  is  to  the 
residance,  as  the  radius  of  the 
axle,  to  the  crank-arm. 

This    machine    is   used    in 


Fi£ 


75. 


\ 


drawing  water  from  wells,  in 
raising  ore  from  mines,  and 
the  like.  It  is  also  used  in  combination  with  other  ma- 
chines. Instead  of  the  crank,  as  shown  in  the  figure,  two 
holes  are  sometimes  bored  at  right  angles  to  each  other 
and  to  the  axis,  and  levers  inserted,  at  the  extremities  of 
which  the  power  is  applied.  The  condition  of  equilibrium 
remains  unchanged,  j^rovided  we  substitute  for  the  crank- 
arm,  the  distance  from  the  point  of  application  of  the 
power  to  the  axis. 

The  Capstan. 

81.  The  Capstan  differs  in  no  material  respect  from  the 
windlass,  except  in  having  its  axis  vertical.  The  capstan 
consists  of  a  vertical  axle  passing  through  guides,  and  hav- 
ing holes  at  its  upper  end  for  the  insertion  of  levers.  It  is 
used  on  shipboard  for  raising  anchors.  The  conditions  of 
equilibrium  are  the  same  as  in  the  windlass. 


The  Differential  Windlass. 

82.  This  differs  from  the  common  windlass  in  having 
its  axle  formed  of  two  cylinders,  A  and  B,  of  different 
diameters.  A  rope  is  attached  to  the  larger  cylinder,  and 
wrapped  several  times  around  it,  after  which  it  passes 
round  the  movable  pulley,  C,  and,  returning,  is  wrapped  in 


100 


MECHANICS. 


a  contrary  direction  about  the  smaller  cylinder,  to  which 
the  second  end  of  the  rope  is  made  fast.  The  power  is 
applied  at  the  crank-handle,  FE,  j, 
and  the  resistance  to  the  hook 
of  the  movable  pulley.  When 
the  crank  is  turned  so  as  to 
wind  the  rope  on  the  larger 
cylinder,  it  unwinds  it  from  the 
smaller  one,  but  in  a  less  de- 
gree, and  the  total  effect  is  to 
raise  the  resistance,  R.  To  de- 
duce the  conditions  of  equilib- 
rium, denote  the  power  by  P,  ^^^' 
the  resistance  by  R,  the  crank-arm  by  c,  the  radius  of  the 
larger  cylinder  by  r,  and  that  of  the  smaller  cylinder  by  r' . 
The  resistance  acts  equally  on  the  two  branches  of  the  rope 
from  which  it  is  suspended,  hence  the  tension  of  each 
branch  may  be  represented  by  \R,  Suppose  the  power 
acts  to  wind  the  rope  on  the  larger  cylinder.  The  moment 
of  the  power  will  be  Pc  ;  the  moment  of  the  tension  of  the 
branch  A  will  be  \Rr\  this  acts  to  assist  the  power ;  the 
moment  of  the  tension  of  the  branch  B  will  be  \Rr,  this 
acts  to  oppose  the  power.  From  the  principle  of  moments, 
we  have. 


Pc  ^  \Rr'  =z  ^Rvy  or,  Pc  =  ^R{r  -  r'); 


whence, 


P  :  R 


r  -  r'  :  2c 


(41) 


That  is,  the  power  is  to  the  i-esistance,  as  the  difference  of 
the  radii  of  the  cylinders,  to  twice  the  cranh-arm. 

By  increasing  the  crank-arm  and  diminishing  the  differ- 
ence between  the  radii  of  the  cylinders,  any  amount  of 
mechanical  advantage  may  be  obtained ;  but  the  amount  of 


ELEMENTARY    MACHINES.  101 

rope  required  for  a  single  turn  is  so  great  as  to  render  ^h^ 
contrivance  in  the  form  described  of  little*  jaradtical  v^Uiei 
This  difficulty  is  avoided  in  a  machiii*  k,no\VRas>Ay5:sTQi?'s 
pulley-block.  In  this  combination,'  tb^re  ^r&  two*  ^Ai^leys 
nearly  equal  in  size,  and  turning  together  as  one  in  the 
upper  block.  An  endless  chain  takes  the  place  of  the  rope, 
and  is  prevented  from  slipping  by  projecting  pins.  The 
power  is  applied  at  the  portion  of  the  chain  that  leaves  the 
larger  pulley,  and  the  chain  continues  to  run  till  the  weight 
is  raised.  To  trace  the  course  of  the  chain,  let  us  com- 
mence at  the  point  where  it  leaves  the  lower  pulley :  from 
this  it  ascends,  passing  around  the  larger  pulley  in  the 
upper  block ;  descending  so  as  to  leave  a  sufficient  amount 
of  slack,  it  again  rises  to  the  upper  block,  passes  around 
the  smaller  pulley,  and  returns  to  the  place  of  beginning. 

Wheel-work. 

83.  The  principle  employed  in  finding  the  relation  be- 
tween the  power  and  resistance  in  a  train  of  wheel- work 
is  the  same  as  that  used  in  dis- 
cussing the  wheel  and  axle  and 
its  modifications.  To  illus- 
trate, we  have  taken  a  case  in 
which  the  powder  is  applied  to 
a  crank-handle  that  is  attached 
to  the  axis  of  a  toothed  wheel, 
A ;  the  teeth  of  this  wheel 
work  into  the  spaces  of  the 
toothed  wheel,  B,  and  the  resistance  is  attached  to  a  rope 
wound  round  the  arbor  of  the  last  wheel.  In  order  that 
A  may  communicate  motion  to  B,  the  number  of  teeth  in 
their  circumferences  should  be  proportional  to  their  radii, 
and  the  spaces  between  the  teeth  in  one  wheel  should  be 


102  MECHANICS. 

large  enough  to  receive  the  teetli  of  tlie  other,  but  not 
large  eno.ugh.  to  allow  much  play.  The  teetli  should 
alwa};s  come  in  coirtact  at  the  same  distances  from  the 
centres  of  the  v/heels^  and  those  distances  are  taken  as  the 
radii  of  the  wheels. 

Denote  the  power  by  P,  the  resistance  by  11,  the  crank- 
arm  by  c,  the  radius  of  the  wheel  A  by  r,  that  of  B  by  r', 
that  of  the  arbor  by  r",  and  suppose  the  power  and  resist- 
ance in  equilibrium.  The  poAver  tends  to  turn  the  wheels 
in  the  direction  of  the  arrow-heads.  This  tendency  is 
counteracted  by  the  resistance  which  tends  to  produce  mo- 
tion in  a  contrary  direction.  If  we  denote  the  pressure  at 
C  by  R',  Ave  have,  from  Avhat  has  preceded, 

Pc  =  Ji'r  and  E'r'  =  Br"; 

whence,  by  multiplication  and  reduction, 

Per'  =  Err",  or,  P  :  P  ::  rr"  :  cr'   ....  (42) 

That  is,  the  j^oiver  is  to  the  resistance,  as  the  coiitmued 
product  of  the  alternate  arms  of  lever,  beginning  at  the 
resista7ice,tothe  continued  product  of  the  alternate  arms  of 
lever  beginning  at  the  power. 

Had  there  been  any  number  of  Avheels  in  the  train  be- 
tween the  poAver  and  resistance,  Ave  should  haA^e  found 
similar  conditions  of  equilibrium. 

Examples. 

1.  A  power  of  5  lbs.,  acting  at  the  circumference  of  a  wheel  whose 
radius  is  5  feet,  supports  a  resistance  of  200  lbs.,  applied  at  the  cir- 
cumference of  the  axle.     What  is  the  radius  of  the  axle  ? 

Ans.  1^  inches. 

2.  The  radius  of  the  axle  of  a  Avindlass  is  3  inches,  and  the  crank- 
arm  15  inches.  What  power  must  be  applied  to  the  crank-handle, 
to  support  a  resistance  of  180  lbs.,  applied  at  the  circumference  of  the 
axle?  ^ws.  36  1bs 


ELEMEJs'TARY    MACHINES.  103 

o.  A  power,  P,  acts  on  a  rope  2  inches  in  diameter,  passing  over  a 
wheel  whose  radius  is  3  feet,  and  supports  a  resistance  of  320  lbs., 
applied  by  a  rope  of  the  same  diameter,  passing  over  an  angle  whose 
radius  is  4  inches.  What  is  the  value  of  P,  the  thickness  of  the  rope 
being  taken  into  account?  Ans.  43-^  lbs. 

The  Screw. 

84.  The  screw  is  a  combination  of  two  inclined  planes 
twisted  round  an  axis.  It  consists  of  a  solid  cylinder, 
enveloped  by  a  spiral  projection  called  the 
thread.  The  thread  may  be  generated  as  fol- 
lows :  let  an  isosceles  triangle  be  placed  so  that 
its  base  shall  coincide  with  an  element  of  the 
cylinder,  and  its  plane  pass  through  the  axis. 
Let  the  triangle  be  revolved  uniformly  about 
the  axis,  and  at  the  same  time  moved  uniformly 
in  the  direction  of  the  axis,  at  such  a  rate  that  it  shall  pass 
over  a  distance  equal  to  the  base  of  the  triangle  in  one 
revolution.  The  solid  generated  by  the  triangle  is  the 
thread  of  the  screw\  The  sides  of  the  triangle  generate 
helicoidal  surfaces,  which  constitute  the  upper  and  lower 
surfaces  of  the  thread.  Each  point  of  these  lines  generates 
a  curve  called  a  helix,  wiiich  is  similar  to  an  inclined  plane 
bent  round  a  cylinder.  The  vertex  generates  the  outer 
helix,  and  the  angular  points  of  the  base  trace  out  the  i7ine?^ 
helix.  The  screw  just  described  has  a  triangular  thread. 
Had  we  used  a  rectangle,  instead  of  a  triangle,  and  imposed 
the  condition,  that  the  motion  in  the  direction  of  the  axis 
during  one  revolution,  should  be  twice  its  base,  we  should 
have  had  a  screw  with  a  rectangular  thread,  as  in  the  figure. 

The  screw  w^orks  into  a  piece  called  a  nut,  generated  in  a 
manner  analogous  to  that  just  described,  except  that  what 
is  solid  in  the  screw  is  hollow^  in  the  nut ;  it  is,  therefore, 
exactly  adapted  to  receive  the  thread  of  the  screws    Some- 


104 


MECHANICS. 


times,  the  screw  is  fast,  and  the  nut  turns  on  it ;  in  this 
case,  the  nut  has  a  motion  of  revolution,  combined  with  a 
longitudinal  motion.  Sometimes^  the  nut  is  fast,  and  the 
screw  turns  within  it;  in  this  case,  the  screw  has  a  motion 
in  the  direction  of  its  axis,  in  connection  with  a  motion 
of  rotation.  The  conditions  of  equilibrium  are  the  same 
for  each.  In  both  cases,  the  power  is  applied  at  the  ex- 
tremity of  a  lever.  We  shall  suppose  the  nut  to  remain 
fast,  and  the  screw  to  be  movable,  and  that  the  resistance 
is  parallel  to  the  axis  of  the  screw.  If  the  axis  is  vertical, 
and  the  resistance  a  weight,  we  may  regard  that  weight  as 
resting  on  one  of  the  helices,  and  sustained  in  equilibrium 
by  a  horizontal  force.  If  the  supporting  helix  be  developed 
on  a  vertical  plane,  by  unrolling  the  surface  of  the  cylinder 
on  which  it  lies,  it  Avill  form  an  inclined  plane,  whose  base 
is  equal  to  the  base  of  the  cylinder  on  which  it  lies,  and 
whose  altitude  is  the  distance  between  the  threads  of  the 
screw. 

Let  AB  be  the  development  of  the  helix,  and  F  the 
force  applied  parallel  to  the  base,  and  iynmediately  to  the 
weight  R,  to  sustain  it  on  the  plane.     We  have,  (Art.  73), 

F  :  R  ::  BC  :  AC. 

But  the  power  is  actually  applied  through  the  medium 
of  a  lever.  Denoting  the  ra- 
dius, OG,  of  the  cylinder  of  the 
supporting  helix,  by  r,  and  the 
arm  of  lever  of  the  power,  P,  by 
p,  we  have,  from  the  principle 
of  the  lever, 

P  :  F  \'.  r  :  p; 
or, 

P  \  F  \\  %'Kr  \  ^ftp.  Fig.  TO. 


"ELEMENTARY   MACHINES.  105 

Combining  this  proportion  with  the  preceding  one,  and 
recollecting  that  AC=  2'rr,  we  deduce  the  proportion, 

P  \  R  -.'.  BC  :^trp (i3) 

That  is,  the  power  is  to  the  resistance,  as  the  distance  be- 
tween the  threads,  to  the  circumference  described  by  the  point 
of  application  of  the  power. 

By  diminishing  the  distance  between  the  threads,  other 
things  being  equal,  any  amount  of  mechanical  advantage 
may  be  obtained. 

The  screw  is  used  for  producing  great  pressures  through 
small  distances,  as  in  pressing  books  for  the  binder,  pack- 
ing merchandise,  expressing  oils,  and  the  like.  On  account 
of  the  great  amount  of  friction,  and  other  hurtful  resist- 
ances developed,  the  modulus  of  the  machine  is  small. 

The  Differential  Screw. 

85.  The  differential  screw  consists  of  an  ordinary  screw, 
into  the  end  of  which  works  a  smaller  screw,  having  its 
axis  coincident  with  the  first.  The  distance  between  the 
threads  of  the  second  screw  is  less  than  that  of  the  first, 
and  this  difference  maybe  made  as  small  as  desirable.  The 
second  screw  is  so  arranged  that  it  admits  of  longitudinal 
motion,  but  not  of  rotation.  By  the  action  of  the  differ- 
ential screw,  the  weight  is  raised  vertically  through  a  dis- 
tance equal  to  the  difference  of  the  distances  between  the 
threads  on  the  two  screws,  for  each  revolution  of  the  point 
of  application  of  the  power. 

Hence,  the  power  is  to  tJie  resistance,  as  the  difference  of 
tlie  distances  between  the  threads  of  the  two  screws  to  the 
circumference  described  by  the  point  of  application  of  the 
power, 

5* 


106 


MECHANICS. 


Fig.  80. 


The  Endless   Screw. 

86.  The  endless  screw  is  ii  screw  secured  by  shoulders, 
so  that  it  cannot  be  moved  longitudinally,  and  working 
into  a  toothed  wheel.  The  dis- 
tance between  the  teeth  is  nearly 
the  same  as  the  distance  between 
the  threads  of  the  screw.  When 
the  screw  is  turned,  it  imparts  a 
rotary  motion  to  the  wheel,  which 
may  bo  utilized  by  any  mechanical 
device.  The  conditions  of  equilib- 
rium are  the  same  as  for  the 
screw,  the  resistance  in  this  case 
being  offered  by  the  wheel,  in  the 
direction  of  its  circumference. 

Machines  of  this  kind  are  used  for  counting  the  number 
of  revolutions  of  an  axis.  An  endless  screw  is  arranged  to 
turn  as  many  times  as  the  axis,  and  being  connected  with 
a  train  of  light  wheel-work,  the  last  piece  of  which  bears 
an  index,  the  number  of  revolutions  can  be  ascertained  at 
any  instant.  For  example,  suppose  the  first  wheel  to  have 
100  teeth,  and  to  bear  on  its  arbor  a  pinion  having  10 
teeth  ;  suppose  this  to  engage  with  another  wheel  having 
100  teeth,  and  so  on.  When  the  endless  screw  has  made 
10,000  revolutions,  the  first  wheel  will  have  made  100  revo- 
lutions, the  second  will  have  made  10  revolutions,  and  the 
third  1  revolution.  By  a  suitable  arrangement  of  indices 
and  dials,  the  exact  number  of  revolutions,  at  any  instant, 
may  be  read  off. 

Examples. 

1.  What  must  be  the  distance  between  the  threads  of  a  screw,  that 
a  power  of  28  lbs.,  acting  at  the  extremity  of  a  lever  25  inclics  Ions:, 
may  sustain  a  weight  of  10,000  lbs.  ?  Ans.  .4896  inches. 


ELEMENTARY   MACHIlfES. 


10? 


2.  The  distance  between  tlie  threads  of  a  screw  is  ^  of  an  inch. 
What  resistance  can  be  supported  by  a  power  of  60  lbs.,  acting  at  the 
extremity  of  a  lever  15  inches  long?  Ans.  16,964  lbs. 

3.  The  distance  from  the  axis  of  tlie  trunions  of  a  gun  weigliing 
2,016  lbs.  to  the  elevating  screw' is  3  feet,  and  the  distance  of  the 
centre  of  gravity  of  the  gim  from  the  same  axis  is  4  inches.  If 
the  distance  between  the  threads  of  the  screw  be  |  of  an  inch,  and 
the  length  of  the  lever  5  inches,  what  power  must  be  applied  to  sus- 
tain the  gun  in  a  horizontal  position  ?  Ans.  4.754  lbs. 


The  Wedge. 

87.  The  wedge  is  a  combination  of  two  inclined  planes. 
It   is   bounded  by  a  rectangle,  BD,  called  the  back;  two 
rectangles,  A  F  and  I) F,  cdlled  faces  ;   and 
two   isosceles   triangles,   called   e7ids.     The 
line,  FF,  in  which  the  faces  meet,  is  the  edge. 

The  power  is  applied  at  the  back,  to  which 
it  should  be  normal,  and  the  resistance  is 
applied  to  the  faces,  and  normal  to  them. 
One  half  the  resistance  is  applied  to  one 
face,  and  the  other  half  to  the  other  face. 
Let  ABC  he  a  section  of  a  wedge  by  a  plane         Fig-  si. 
at  right  angles  to  the  edge.    Denote  the  power  by  P,  the  re- 
sistance opposed  to  each  face  by  ^R,  and 
the  angle  BA  C  by  2(p.    Produce  the  direc- 
tions of  the  resistances  till  they  intersect 
in  0.     This  point  will  be  on  the  line  of 
the  direction  of  the  power.     Because  the 
three  forces  P,  ^R,  and  ^R  are  in  equilib- 
rium, we  have,  (Art.  33), 
F  :  iR  ::  sinFOD  :  sinPOD  ...  (44) 

But,  DO  and  FO  are  perpendicular  to 
^Cand  AB  :  hence. 


sinFOD  =  sin2gp  =  2sm:p  cos?. 


108  MECHANICS. 

In  like  manner,  PO  and  DO  are  perpendicular  to  KG 
and  AC ;  hence, 

sinPOi>  =  sin  J  CK  =  costp. 

Substituting,  and  reducing,  we  have, 

P  :  \R  w  2sin:p  :  1, 

or,  P  :  R  ::  KC  '.AC (45) 

That  is,  the  poiver  is  to  the  resistance,  as  half  the  breadth 
of  the  back,  is  to  the  length  of  the  face. 

The  mechanical  advantage  of  the  wedge  may  be  in- 
creased by  diminishing  the  breadth  of  the  back,  or,  in 
other  words,  by  making  the  edge  sharper.  The  principle 
of  the  wedge  finds  an  application  in  cutting  instruments. 
By  diminishing  the  thickness  of  the  back,  the  instrument 
is  weakened;  hence  the  necessity  of  forming  cutting  instru- 
ments of  hard  and  tenacious  material. 

Application  of  the  Principle  of  Virtual  Moments. 

88.  The  preceding  conditions  of  equilibrium  might  have 
been  deduced  from  the  principle  of  virtual  moments.  To 
illustrate  the  mode  of  proceeding,  let  us  take  the  case  of  a 
single  movable  pulley,  and  suppose  P  and  R  to  be  in  equi- 
librium. Let  the  machine  be  set  in  motion  until  P  has 
acted  through  a  very  small  distance,  FG,  in  its  ^ 
own  direction ;  the  force,  R,  will  have  acted 
in  the  same  time  through  some  distance,  DE, 
contrary  to  its  own  direction.  From  the  prin- 
ciple of  virtual  moments,  we  have, 
P  X  FG  -  R  X  DE  =  0. 

In  order   that  R  may  act  through  a  dis- 


y 


"n 


tance,  DE,  each  branch  of  the  rope  must  be  ^ 

shortened    by  an    equal   amount;    in   other        Fig.  S3. 


ELEMENTARY    MACHINES.  109 

words,  tlie  force,  P,  must  act  through  twice  the  distance, 
DE.     Making  FG  =  "ZDE,  and  reducing,  we  have, 

P  -  ii2, 

as  already  shown.     In  like  manner,  the  conditions  of  equi- 
librium for  other  machines  may  be  deduced. 

Hurtful  Resistances. 

89.  The  principal  hurtful  resistances  that  must  be  taken 
into  account  in  modifying  the  relations  between  the  power 
and  resistance,  are  friction,  adliesion,  stiffness  of  cords, 
and  at^nospheric  resistance. 

Friction. 

90.  Friction  is  the  resistance  one  body  experiences  in 
moving  on  another,  the  two  being  pressed  together  by  some 
force.  This  resistance  arises  from  inequalities  in  the  sur- 
faces, the  projections  of  one  sinking  into  the  depressions 
of  the  other.  In  order  to  overcome  this  resistance,  suffi- 
cient force  must  be  applied  to  break  off,  or  bend  down,  the 
projecting  points,  or  else  to  lift  the  moving  body  clear  of 
them.  The  force  thus  applied,  is  equal,  and  directly 
opposed  to  the  force  of  friction,  which  is  tangential  to  the 
two  surfaces.  The  force  that  presses  the  surfaces  together, 
is  normal  to  both  at  the  point  of  contact. 

Between  certain  bodies,  friction  is  somewhat  different 
when  motion  is  just  beginning,  from  what  it  is  when  mo- 
tion has  been  established.  The  friction  developed  when  a 
body  is  passing  from  a  state  of  rest  to  a  state  of  motion,  is 
called 'friction  of  quiescence;  that  between  bodies  in  mo- 
tion, is  called  friction  of  motion. 

The  following  Iciws  of  friction  have  been  established  by 
experiment,  viz.: 


110  MECHANICS. 

First,  friction  of  quiescence  hetwecn  the  same  bodies,  is 
proportional  to  the  nor 7nal  pressure,  and  independent  of  the 
extent  of  the  surfaces  in  contact. 

Secondly,  friction  of  motion  between  the  same  bodies,  is 
proportional  to  the  normal  pressure,  and  independent,  both 
of  the  extent  of  surface  of  contact,  and  of  the  velocity  of  the 
moving  body. 

Thirdly, /or  compressible  bodies,  friction  of  quiescence  is 
greater  than  friction  of  motion :  for  bodies  which  are  in- 
compressible, the  difference  is  scarcely  apjireciable. 

Friction  may  be  diminished  by  the  interposition  of 
unguents,  which  fill  up  the  cavities,  and  so  diminish  the 
roughness  of  the  rubbing  surfaces.  For  slow  motions  and 
great  pressures,  the  more  substantial  unguents  are  used, 
such  as  lard,  tallow,  and  certain  mixtures ;  for  rapid  mo- 
tions, and  light  pressures,  oils  are  generally  employed. 

Methods  of  finding  the  Coefficient  of  Friction. 

91.  The  quotient  obtained  by  dividing  the  force  of  fric- 
tion by  the  normal  pressure,  is  called  the  coefficient  of  fric- 
tion ;  its  value  for  any  two  substances,  may  be  determined 
as  follows : 

Let  ^5  be  a  horizontal  plane  formed  of  one  of  the  sub- 
stances, and  0  a  cubical  block  of  the  •ther.  Attach  a 
string,  OC,  to  the  block,  so  that 
its  direction  shall  pass  through 
the   centre   of  gravity,   and  be 


"7 


^ 


^zy 


parallel   to  AB;  let  the  string 

pass  over  a  fixed  pulley,  C,  and  ^  J»P 

let  a  weight,  F,  be  attached  to  its  Fig.  84. 

extremity. 

Increase  i^  till  O  just  begins  to  slide  along   the  plane, 
then  will  F  be  the  force  of  friction.     Denote  the  normal 


ELEMENTARY    MACHIXES.  Ill 

pressure,  by  P,  and  the  coefficient  of  friction,  by/.    From 
the  definition,  we  have, 

J       p- 

In  this  manner,  values  for  /  may  be  found  for  different 
substances,  and  arranged  in  tables. 

The  value  of  /,  for  any  substance,  is  the  uriit,  or  coeffi- 
cient of  friction.  Hence,  we  may  define  the  unit,  or  coeffi- 
cient of  friction,  to  be  the  friction  due  to  a  normal  pressure 
of  one  jponnd. 

Having  the  normal  pressure  in  pounds,  and  the  coeffi- 
cient of  friction,  the  entire  friction  may  be  found  by  mul- 
tiplying these  quantities  together. 

There  is  a  second  method  of  finding  the  value  of  /,  as 
follows : 

Let  ^^  be  an  inclined  plane,  formed  of  one  of  the  sub- 
stances, and  0  a  block,  of  the  other.  Elevate  the  plane  till 
the  block  just  begins  to  slide  down 
by  its  own  weight.  Denote  the  incli- 
nation, at  this  instant,  by  a,  and  the 
weight  of  0,  by  W.  Resolve  irinto 
two  components,  one  normal  to  the 
plane,  and  the  other  parallel  to  it. 
Denote  the  former  by  P,  and  the  latter  by  Q.  Since  0  W 
is  perpendicular  to  A  C,  and  OP  to  AB,  the  angle,  WOP, 
is  equal  to  a.     Hence, 

P  =  Wcos%,  aud  Q  =  Wsina. 
The  normal  pressure  being  equal  to  Ifcosa,  and  the  force 
of  friction  being  Tfsina,  we  shall  have,  from  the  principle 
already  explained, 

.      Wsina 

f=  ^fj? ?  01"'  f=  tana. 

^cosa' 


112  MECHANICS. 

The  angle  a  is  called  the  angle  of  friction. 
The  values  of  /,  in  some  of  the  more  common  cases,  are 
given  in  the  following 

TABLE. 

Bodies  between  which  friction  takes  jilace.  Coefficient  of  friction. 

Iron  on  oak 62 

Cast-iron  on  oak 49 

Oak  on  oak,  fibres  parallel 48 

Do.,  do.,  greased 10 

Cast-iron  on  cast-iron .15 

Wrought-iron  on  wrought-iron 14 

Brass  on  iron 16 

Brass  on  brass 20 

Wrought-iron  on  cast-iron 19 

Cast-iron  on  elm 19 

Soft  limestone  on  the  same. 64 

Hard  limestone  on  the  same 38 

Influence  of  Friction  on  an  Inclined  Plane. 
92.  To  show  the  manner  of  taking  account  of  friction, 
let  us  consider  the  case  of  a  body  sliding  on  an  inclined 
plane.      Let  ^^  be  the  plane,    0 
the  body,  P  the    power,    situated 
in  a  plane  perpendicular  both  to 
the  horizon  and  to  the  given  plane, 
and  suppose  the  body  on  the  eve 
of  motion  up  the  plane.     Denote 
the  weight  of  the  body  by  i?,  the 
inclination     of    the    plane   by   a,  ^^^-  *• 

and  the  angle  between  the  power  and  the  normal  to  the 
plane  by  /3.  Let  }*  and  J^  be  resolved  into  components 
parallel  and  ])erpendicular  to  the  plane.  We  have,  for  the 
parallel  components,  i^sina  and  Psin,^,  and  for  the  perpen- 


.       ELEMENTARY    MACHINES.  113 

clicuiar  components,  7?cosa  and  FcosS.  The  resultant  of 
the  normal  components  is  Ecosol  —  Pcos.^^ ;  and  the  force 
of  friction  (Art.  91)  is  equal  to 

/(i^cosa  —  Fcos^). 

Because  the  body  is  on  the  eve  of  motion  up  the  plane, 
the  component  Psin/3  must  be  equal  and  directly  opposed 
to  the  resultant  of  the  force  of  friction  and  the  component 
i?sina ;  hence,  we  must  have, 

Psin/3  =  i^sina  +/(i?cosa  —  PcosS). 

Performing  the  multiplications  indicated,  and  reducing, 
we  have, 

P^H.  ^'"«+/°^^ (46) 

amis  -\-fcosl3  ^     ' 

If  an  equilibrium  exist,  the  body  being  on  the  eve  of 
motion  down  the  plane,  we  have, 

Psin3  -j- /(Rcosa  —  PcosiS)  =  i?sina. 
AVhence,  by  reduction, 

When  a,  /3,  and/,  are  given,  P  may  be  found  in  terms 

of  P. 

Example. 

Let  the  plane  be  of  oak,  the  sliding  body  of  cast-iron,  the  inclina- 
tion of  the  plane  to  the  horizon  20°,  and  the  angle  between  the  power 
and  a  normal  to  the  given  plane  64°.  Required  the  relation  between 
P  and  i?,  when  the  body  is  on  the  eve  of  motion. 

We  have,  /  =  .49 ;  sin  a  =  .34;  cos  «  =  .94 ;  sin  /?  =  90';  and 
cos  /3  =  .44.  Substituting,  in  (46)  and  (47),  and  reducing,  we  have,  in 
the  former,  P  =  .71  B,  and  in  the  latter,  P  =  .38  R. 

liimiting  Angle  of  Resistance. 

93.  Let  AB  be  a  plane,  and  0  a  body  resting  on  it.  Let 
E  be  the  resultant  of  all  the  forces  acting  on  it,  including 


114 


MECHANICS. 


its  weight.  Denote  the  angle  between 
R  and  the  normal  to  AB,  by  a,  and 
suppose  R  to  be  resolved  into  two 
components,  P  and  Q,  the  former 
parallel  to  AB,  and  the  latter  per- 
pendicular to  it ;  we  have, 

P  =  i?sina,  and  Q  =  7?cosa. 

The  friction  due  to  the  normal  pressure  is  equal  to 
fRcosa..  When  the  tangential  component  i^sina  is  less 
than  /i?cosa,  the  body  will  remain  at  rest;  w^hen  it  is 
greater  than  fRcosa,  the  body  will  slide  along  the  plane  ; 
and  wiien  the  two  are  equal,  tlie  body  will  be  in  a  state 
bordering  on  motion  along  the  plane.  Placing  the  two 
equal,  we  have, 

fRcosa  =  i?sina  ;     .-.     tana  =/. 

This  value  of  a  is  called  the  limiting  angle  of  resistance, 
and  is  equal  to  the  inclination  of  the  plane,  when  the  body 
is  about  to  slide  down  by  its  own  weight. 
If  OR  be  revolved  about  the  normal,  it 
will  generate  a  conical  surface,  called  the 
limiting  cone  of  resistance.  If  the  re- 
sultant of  all  the  forces  acting  on  0,  lie 
within  this  cone,  the  body  will  remain  at 
rest ;  if  it  lie  without,  the  body  will  move  along  the  plane 
in  the  direction  determined  by  a  plane  through  the  force 
and  the  normal ;  if  it  lie  on  the  surface  of  the  cone,  the 
body  will  be  on  the  eve  of  motion  along  the  plane  in  a 
direction  determined  as  before.  Tlie  last  principle  is  appli- 
cable in  many  cases,  and  may  be  enunciated  as  follows: 
Wlien  one  body  is  on  the  eve  of  sliding  along  another,  the 
resultant  of  all  the  forces  acting  on  the  former,  including 
i*s  weight,  makes  an  angle  with  the  normal  to  the  surfaces 


ELEMENTARY    MACHINES.  115 

at  their  point  of  contact  equal  to  the  angle  of  friction  of  the 
two  bodies. 

Friction  on  an  Axle. 

94.  The  principle  demonstrated  in  the  last  article  enables 
us  to  determine  the  position  of  equilibrium  of  a  horizontal 
axle  revolving  in  a  cylindrical  box. 

Let  0'  be  the  centre  of  the  cross  section  of  the  axle,  and 
0  that  of  the  box,  and  let  N  be  their  point  of  contact 
when  the  power  is  on  the  point  of  overcom- 
ing friction.  At  N  let  NT  be  drawn  tan- 
gent to  both  circles.  The  axle  may  noAv 
be  regarded  as  a  body  resting  on  the  inclined 
plane,  NT,  and  on  the  eve  of  sliding  along 
it.  Hence,  the  resultant  of  all  the  forces 
acting  on  the  axle,  except  friction,  must 
pass  through  N,  and  make  an  angle  with  NO  equal  to  the 
angle  of  friction  between  the  axle  and  box.  If  the  axle  be 
rolled  further  up  the  side  of  the  box,  it  will  slide  back  to 
iV^;  if  it  be  thrust  down  the  box,  it  will  roll  back  to  N. 
If  all  the  forces  acting  on  the  axle,  except  friction,  arc  ver- 
tical, NT  will  make  with  the  horizon  an  angle  equal  to 
that  of  friction.  In  this  case  the  relation  between  the 
power  and  resistance  may  be  found,  as  in  Art.  92. 

Line  of  Least  Traction. 

95.  The  force  employed  to  draw  a  body  uniformly  along 
an  inclined  plane,  is  called  the  force  of  traction  ;  and  the 
direction  of  this  force  is  the  line  of  traction.  In  equation 
(46),  P  is  the  force  of  traction,  and  /3  is  the  angle  the  line 
of  traction  makes  with  the  normal.  AYhen  /3  varies,  other 
things  being  equal,  the  value  of  P  also  varies;  there  is  evi- 
dently some  value  of  /3  that  will  render  P  least  possible : 


116  MECHANICS. 

the  direction  of  P,  in  this  case,  is  the  line  of  least  traction ; 
it  is  along  this  line  that  a  force  can  be  applied  with  great- 
est advantage,  to  draw  a  body  up  an  inclined  plane.  If  we 
examine  the  expression  for  P,  in  equation  (46),  we  see  that 
the  numerator  is  constant;  therefore,  the  expression  for 
P  will  be  least  possible  when  the  denominator  is  greatest 
possible.  By  a  simple  process  of  the  Diflferential  Calculus, 
it  may  be  shown  that  the  denominator  will  be  greatest 
possible,  or  a  maximum,  when, 

/  =  cot/3,    or,  /  =  tan(90°  -  /3). 

That  is,  the  power  will  be  applied  most  advantageously, 
when  it  makes  an  angle  with  the  inclined  plane  equal  to 
the  angle  of  friction. 

From  the  second  value  of  P,  it  may  be  shown,  in  like 
manner,  that  a  force  will  be  most  advantageously  applied, 
to  prevent  a  body  from -sliding  down  a  plane,  when  its 
direction  makes  an  angle  with  the  plane  equal  to  the  sup- 
plement of  tlie  angle  of  friction,  the  angle  being  estimated, 
as  before,  from  that  part  of  the  plane  lying  above  the  body. 

Resistance  to  Rolling. 

96.  Resistance  to  rolling,  sometimes  called  rolling  fric- 
tion, is  the  resistance  experienced  when  one  body  rolls  on 
another,  the  two  being  pressed  together  by  some  force.  It 
arises  from  inequalities  in  the  two  surfaces,  and  also  from 
distortion  caused  by  the  force  that  presses  the  bodies  to- 
gether. The  coefficient  is  the  quotient  obtained  by  dividing 
the  entire  resistance  by  the  normal  pressure. 

The  following  laws  have  been  established,  when  a  cylin- 
drical body  rolls  on  a  plane : 

First,  the  friction  is  proportional  to  the  normal  pre^ 

sure. 


ELEMENTARY    MACHINES.  117 

Secondly,  it  is  inversely  proportional  to  the  diameter  of 
the  cylinder  or  wheel. 

Thirdly,  it  increases  as  the  surface  of  contact  and  velocity 
increase. 

In  many  cases  there  is  a  comhination  of  both  sliding 
and  rolling  friction  in  the  same  machine.  Thus,  in  a  car 
on  a  railroad  track,  the  friction  at  the  axle  is  sliding,  and 
that  between  the  wheel  and  track  is  rolling. 

Work  of  Friction. 

97.  The  work  of  friction  is  equal  to  the  work  of  the 
force  necessary  to  overcome  it.  It  is  therefore  measured  by 
the  product  of  the  force  of  friction  into  the  path  through 
which  it  is  exerted.  In  case  of  an  axle  revolving  in  a  box, 
the  work  during  one  revolution  is  equal  to  the  force  of 
friction  multiplied  by  the  circumference  of  the  axle. 

Adhesion. 

98.  Adhesion  is  the  resistance  one  body  experiences  in 
moving  on  another  in  consequence  of  cohesion  between 
the  molecules  of  the  surfaces  in.  contact.  This  resist- 
ance increases  when  the  surfaces  are  allowed  to  remain  in 
contact  for  some  time,  but  is  very  slight  when  motion  has 
been  established.  Both  theory  and  experiment  show  that 
adhesion  between  the  same  surfaces,  is  proportional  to  the 
extent  of  the  surface  of  contact. 

The  coefficient  of  adhesion  is  obtained  by  dividing  the 
entire  adhesion  by  the  area  of  the  surface  of  contact.  De- 
noting the  entire  adhesion  by  A,  the  area  of  the  surface  of 
contact  by  S,  and  the  coefficient  of  adhesion  by  a,  we  have, 

a  =  — ,    or,    A  =  aJ^. 

o 

To  find  the  entire  adhesion,  multiply  the  unit  of  adhe- 
sion bv  the  area  of  the  surface  of  contact. 


118  MECHANICS. 

Stiffness  of  Cords. 
99.  Let  0  be  a  pulley,  with  a  cord,  AB ,  wrapped  round 
its  circumference ;  and  suppose  a  force,  P,  applied  at  B, 
to  overcome   a   resistance,  R.     As  the  rope 
winds  on  the  pulley,  at  C,  its  rigidity  acts  to 
increase  the  arm  of  lever  of  B,  and  to  over- 
come  this   rigidity   an    additional    force    is 
required.     This  additional  force  may  be  re- 
presented by  the  expression. 


j/a  +  hB\ 


in  which  d  depends  on  the  character  and  size  of  the  rope, 
a  on  its  natural  rigidity,  bB  on  the  rigidity  due  to  the  load, 
and  D  is  the  diameter  of  the  wheel.  The  values  of  d,  a, 
and  i  have  been  found  by  experiment  for  different  kinds 
of  rope,  and  tabulated. 

Atmospheric  Resistance. 

100.  The  atmosphere  offers  a  resistance  to  bodies  moving 
through  it,  in  consequence  of  the  inertia  of  its  particles. 
For  the  same  extent  of  surface  the  resistance  varies  as  the 
square  of  the  velocity.  For,  if  the  velocity  be  doubled,  twice 
as  many  particles  will  be  met  with  in  a  given  time,  and  each 
particle  will  be  impinged  against  by  the  moving  body  with 
twice  the  force ;  nence,  the  resistance  will  be  quadrupled. 
In  a  similar  manr  er  it  may  be  shown  that  if  the  velocity 
be  tripled,  the  resistance  will  be  nine  times  as  great,  and  so 
on.  If,  therefore,  the  resistance  on  a  square  foot  of  surface 
be  determined  for  a  given  velocity,  the  resistance  offered  to 
any  surface,  and  for  any  volocity,  may  be  computed. 

For  the  detailed  methods  of  taking  hurtful  resistances 
into  account,  the  reader  is  referred  to  more  extended 
treatises  on  practical  mechanics. 


CHAPTER  V. 

RECTILINEAR   AND    PERIODIC   MOTIOK. 
Motion. 

101.  A  point  is  in  motion  when  it  continually  changes 
its  position  in  space.  When  the  path  of  the  moving  point 
is  a  straight  line,  the  motion  is  rectilineal' ;  when  it  is  a 
curved  line,  the  motion  is  curvilinear.  When  the  motion 
is  curvilinear,  we  may  regard  the  path  as  made  up  of  infi- 
nitely short  straight  lines ;  that  is,  we  may  consider  it  as  a 
polygon,  whose  sides  are  infinitely  small.  If  any  side  of  this 
polygon  be  prolonged  in  the  direction  of  the  motion,  it  will 
be  tangent  to  the  curve.  Hence,  we  say,  that  a  point  moves 
in  the  direction  of  a  tangent  to  its  path. 

Uniform  Motion. 

102.  Uniform  motion  is  that  in  which  the  moving  point 
describes  equal  spaces  in  any  equal  portions  of  time.  If 
we  denote  the  space  passed  over  in  one  second  by  v,  and  in 
t  seconds  by  s,  we  have,  from  the  definition, 

S  z=  Vtj        .'.     V  —  — 
t 

From  the  first  of  these  equations,  we  see  that  the  space 
described  in  any  time  is  equal  to  the  product  of  the  velocity 
and  time ;  from  the  second,  we  see  that  the  velocity  is  equal 
to  the  space  described  in  any  time,  divided  hy  that  time. 

If  the  moving  point  had  passed  over  a  space  s,  at  the 


120 


MECHANICS. 


beginning  of  the  time  t,  the  relation  between  the   spaces 
and  times  would  be  given  by  the  equation, 

s  =  s'  ■]-  vt (48) 

In  this  equation,  ^9'  is  called  the  initial  space. 

Uniformly  Varied    Motion. 

103.  IJiiiformly  varied  motion,  is  that  in  which  the 
^Telocity  increases  or  diminishes  uniformly.  In  the  former 
case,  the  motion  is  accelerated,  in  the  latter,  retarded.  In 
both  the  moving  force  is  constant. 

To  find  the  relation  between  the  spaces  passed  over,  and 
the-  velocities  generated,  m  any  time,  let  the  acceleration 
due  to  the  moving  force,  (Art.  18),  be  denoted  by/,  and  the 
velocity  generated  in  t  seconds  by  v.  The  acceleration  is 
the  velocity  generated  in  one  second,  and  because  the  velo- 
city generated  is  proportional  to  the  time,  we  have,  from 
the  definition, 

V  =ft (49) 

Because  the  velocity  increases  uniformly,  the  space  de- 
scribed in  any  time  is  the  same  as  though  the  body  had 
moved  uniformly  during  that  time,  with  its  mean,  or  average 
velocity.  At  the  beginning  of  the  time  t,  the  velocity  is  0, 
at  the  end  of  that  time  it  is//;  hence,  the  average  velocity 
during  the  time  t  is  ^ft ;  multiplying  this  by  the  time  /, 
we  have,  for  the  space  described,  ^ft  X  t,  or,  denoting  the 
space  by  s,  we  have, 

.^•  =  i//= (50) 

Equations  (49)  and  (50)  express  the  circumstances  of 
motion  of  a  body  moving  from  a  state  of  rest,  under  the 
action  of  a  constant  force :  from  the  former  we  see  that 
the  velocities  are  proportional  to  the  times,  and  from  the  lat- 


RECTILINEAR    AND    PERIODIC    MOTION.  121 

icr  we  see  that  the  spaces  are  proportional  to  the  squares  of 
the  times. 

If  in  equation  (50)  we  make  ^  =  1,  we  find, 


=  if;  or,/=2s. 


That  is,  if  a  body  move  from  rest,  under  the  action  of  a 
constant  force,  the  acceleration  is  measiired  hy  twice  the 
space  passed  over  in  the  first  second. 

It  follows,  from  the  principle  of  inertia,  that  the  velocity 
generated  in  any  time  is  entirely  independent  of  the  state 
of  the  body  at  the  beginning  of  that  time.  If,  therefore, 
the  body  has  a  velocity  v'  at  the  beginning  of  the  time  t, 
equation  (49)  will  become 

v  =  v'  +ft (51) 

In  this  equation  v'  is  called  the  initial  velocity. 

If  we  suppose  the  body  to  have  passed  over  a  space  s', 
called  the  i?iitial  space,  before  the  beginning  of  t,  the  final 
space  will  be  made  up  of  three  parts ;  first,  the  initial  space, 
s';  second,  the  space  due  to  the  initial  velocity  v',  and  equal 
to  v't;  third,  the  space  due  to  the  action  of  the  constant 
force/  during  the  time  t,  equal  to  ^ff.     Hence, 

s  =  s'-h  v't  +  iff (52) 

Equations  (51),  and  (52),  may  be  made  to  conform  to  any 
case  of  uniformly  varied  motion,  by  giving  suitable  values 
to  s',  v',  and  /;  it  is  to  be  observed,  that  any  one  of  these 
quantities  may  be  either  plus  or  minus.  When  f  is  essen- 
tially positive  the  motion  is  accelerated,  when  /  is  essen- 
tially negative  the  motion  is  retarded. 

Application  to  Falling  Bodies. 

104.  The  force  of  gravity  is  the  force  exerted  by  the 
earth  on  all  bodies  exterior  to  it.     It  is  found,  by  observa- 

6 


122  MECHANICS. 

tion,  that  this  force  is  directed  tmvard  the  centre  of  the 
earth,  and  that  its  intensity  varies  inversely,  as  the  square 
of  the  distance  from  the  centre. 

Because  the  centre  of  the  earth  is  so  distant  from  the 
surface,  the  variation  in  intensity  for  small  elevations  above 
the  surface  is  inappreciable.  Hence,  we  may  regard  the 
force  of  gravity  at  any  place  on,  or  near,  the  earth's  surface, 
as  constant ;  in  which  case,  the  equations  of  the  preceding 
article  are  applicable.  The  force  of  gravity  acts  equally 
on  all  the  particles  of  a  body,  and  were  there  no  resistance 
offered,  it  would  impart  the  same  velocity,  in  the  same 
time,  to  any  two  bodies  whatever.  The .  atmosphere,  how- 
ever, offers  a  resistance,  which  tends  to  retard  the  motion 
of  bodies  falling  through  it ;  and  of  two  bodies  of  equal 
mass,  it  retards  that  one  most,  which  presents  the  greatest 
surface  to  the  direction  of  the  motion.  In  discussing  the 
laws  of  falling  bodies,  it  will  be  found  convenient  to  re- 
gard them  as  being  in  a  vacuum,  and  in  this  case  the 
equations  of  the  preceding  article  are  immediately  applica- 
ble. The  effects  of  atmospheric  resistance  maybe  taken 
into  account,  as  corrections,  or  in  certain  cases  the  mo- 
tions may  be  made  so  slow  that  their  effects  may  be  neg- 
lected. 

If  we  denote  the  acceleration  due  to  gravity  by  g,  and 
the  space  fallen  through  by  h,  both  being  regarded  as  posi- 
tive downward,  we  have,  from  (49)  and  (50), 

v  =  gt     (53) 

h  =  yf (54) 

That  is,  the  velocities  are  proportional  to  the  times,  and 
the  spaces  to  the  squares  of  the  thnes. 

The  value  of  g  in  the  latitude  of  New  York  is  not  far 
from  32^  feet;  making  g  =  32|  feet,  and  giving  to  t  the 


RECTILINEAR    AND    PERIODIC    MOTION. 


123 


values  1',  2",  3%  &c.,  in  equations  (53)  and  (54),  we  have 
the  results  given  in  the  following 

TABLE. 


TIME    ELAPSED. 

VELOCITIES    ACQUIRED. 

SPACES   DESCRIBED. 

Seconds. 

Feet. 

Feet. 

1 

d2i 

16tV 

2 

6^ 

64J 

3 

96^ 

144| 

4 

128| 

257i 

5 

160|- 

402,V 

&c. 

&c. 

&c. 

Solving  equation  (54)  with  respect  to  t,  we  have, 

.  .  (55) 


^    9 


That  is,  the  number  of  seconds  required  for  a  hody  to  fall 
through  any  height  is  equal  to  the  square  root  of  the  quotient 
obtained  by  dividing  twice  the  height  in  feet  by  32|. 

Substituting  this  value  of  t,  in  equation  (53),  we  have, 

v  —  gy—,  or  v"  —  2gh ; 
whence,  by  solving  with  respect  to  v,  and  h, 


and    h  = 


(56) 


In  these  equations,  v,  is  called  the  velocity  due  to  the 
height  h,  and  h,  the  height  due  to  the  velocity  v. 

If  the  body  be  projected  downward  with  a  velocity  v', 
the  circumstances  of  motion  will  be  made  known  by  the 
equations, 

V  =v'  -\-  gt, 

h  =  v't  +  yt\ 


124  MECHANICS. 

Ill  these  equations,  the  origin  of  spaces  is  at  the  point 
from  which  the  body  is  projected  downward. 

Motion  of  Bodies  projected  vertically  upward. 

105.  Suppose  a  body  projected  vertically  upward  from 
the  origin  of  spaces,  with  a  velocity  v',  and  afterward  acted 
on  by  the  force  of  gravity.  In  this  case,  the  force  of  grav- 
ity acts  to  retard  the  motion.  Making,  in  (51)  and  (52), 
s'  =  0,  f  =  —  g,  and  s  =  h,  they  become, 

v  =  v'  -gt (57) 

h  =  v't-  \gf  ....  (58) 

In  these  equations  h  is  positive  upward,  and  negative 
downward. 

From  equation  (57),  we  see  that  the  velocity  diminishes 
as  the  time  increases.     The  velocity  is  0,  when, 

v' 
v'  —  gt  =  0,  or,  when  t  —  —. 

v' 
When  t  is  greater  than  — ,  v  is   negative,  and  the  body 

retraces  its  path :  hence,  the  time  required  for  the  tody  to 
reach  its  highest  elevation,  is  equal  to  the  initial  velocity, 
divided  hy  the  force  of  gravity. 

Eliminating  t,  from  (57)  and  (58),  we  have, 

^  =  T '''' 

Making  t;  =  0,  in  the  last  equation,  we  have, 

^=S ('"> 

Hence,  the  greatest  height  to  ivhich  the  body  will  ascend, 
is  equal  to  the  square  of  the  initial  velocity,  divided  hy  twice 
the  force  of  gravity. 


RECTILINEAR   AND    PERIODIC    MOTION.  125 

This  height  is  thafc  due  to  the  initial  velocity,  (Art.  104). 

v' 
If,  in  (57),  we  make  t= ^',  we  find, 

v  =  gt' (61) 

v' 
If,  in  the  same  equation,  we  make  t  =  —  +  ^',  we  find, 

v=  -gt' (62) 

Hence,  the  velocities  at  equal  times  before  ayid  after  reacli- 
imj  the  highest  points  are  equal. 

Tha  difference  of  signs  shows  that  the  body  is  moving  in 
opposite  directions  at  the  times  considered. 

If  we  substitute  these  values  of  v  successively,  in  (59), 
we  find  in  both  cases, 

7      v"  -  g"t" 

^g 

hence,  the  points  at  which  the  velocities  are  equal,  in 
ascending  and  descending,  are  equally  distant  from  the 
highest  point ;  that  is,  they  are  coincident.  Hence,  if  a 
body  be  ])rojected  vertically  upward,  it  ivill  ascend  to  a  cer- 
tain point,  and  then  return  upon  its  jJ^th,  in  such  manner, . 
that  the  velocities  in  ascending  and  descending  are  equal  at 
the  same  points. 

Examples. 

1.  Through  what  distance  will  a  body  fall  from  rest  in  a  vacuum, 
in  10  seconds,  and  through  what  space  will  it  fall  during  the  last 
second  ?  Am.  1608^?  ft.,  and  305^  ft. 

3.  In  what  time  will  a  body  fall  from  rest  through  1200  feet? 

Ans.  8.63  sec. 

3.  A  body  was  observed  to  fall  through  a  height  of  100  feet  in  the 
last  second.  How  long  was  the  body  falling,  and  through  what  dis- 
tance did  it  descend  ? 


126  MECHANICS. 

SOLUTION. 

If  we  denote  the  distance  by  h,  and  the  time  by  t,  we  have, 
h  =  ^gt"^  and  ^  -  100  =  ^^g{t  -  If  • 
.:  t  =  8.6  sec,  and  h  =  203.44  ft.  Ans. 

4.  A  body  falls  through  300  feet.  Through  what  distance  does  it 
fall  in  the  last  two  seconds  ? 

The  entire  time  occupied,  is  4.32  seconds.  The  distance  fallen 
through  in  2.32  sec,  is  86.57  ft.  Hence,  the  distance  required  is 
300  ft.  -  86.57  ft.  =  213.43  ft.  Ans. 

5.  A  body  is  projected  upward,  witli  a  velocity  of  60  feet.  To 
what  height  will  it  rise?  Ans.  55.9  ft. 

6.  A  body  is  projected  upward,  with  a  velocity  of  483  ft.  In  what 
time  will  it  rise  1610  feet? 

We  have,  from  equation  (58), 

1610  =  483^  -  16  hi' ;       .'•  t  =  V9¥  ±  ¥^ ; 
or,  t  =  26.2  sec,  and  t  =  3.82  sec. 

The  smaller  value  of  t  gives  the  time  required ;  the  larger  value 
gives  the  time  occupied  in  rising  to  its  greatest  height,  and  returning 
to  the  point  1610  feet  from  tlie  starting  point. 

7.  A  body  is  projected  upward,  with  a  velocity  of  161  feet,  from 
a  point  214|  feet  above  the  earth.  In  what  time  will  it  reach  the 
earth,  and  with  what  velocity  will  it  strike? 

SOLUTION. 

The  body  will  rise  402.9  ft.  The  time  of  rising  will  be  5  sec ;  the 
time  of  falling  to  the  earth  will  be  6.2  sec.  Hence,  the  required  time 
is  11.2  sec.     The  required  velocity  is  199  ft. 

8.  Suppose  a  body  to  have  fallen  through  50  feet,  when  a  second 
begins  to  fall  just  100  feet  below  it.  How  far  will  the  latter  body  fall 
before  it  is  overtaken  by  the  former  ?  Ajis.  50  feet. 

Restrained  Vertical  Motion. 

106.  We  have  seen,  (Art.  18),  that  the  acceleration  dtte 
to  a  moving  force  is  equal  to  the  moving  force  divided  hy  the 
mass  inoved.  Hence,  in  the  case  of  a  body  falling  freely, 
the  moving  force  varies  directly  as  the  mass  moved,  and 
the  acceleration  is  constant.     If,  however,  we  increase  the 


RECTILINEAR   AND    PERIODIC   MOTION.  127 

mass  moved,  without  changing  the  moving  force,  we  shall 
correspondingly  diminish  the  acceleration;  and  in  this 
manner  we  may  render  it  as  small  as  possible.  This  result 
may  be  attained  by  the  combination  repre-  ^— ^ 
sented  in  the  figure.  In  it,  ^  is  a  fixed  pulley, 
mounted  on  a  horizontal  axis,  W  and  W  are 
unequal  weights  attached  to  the  extremities  of 
a  flexible  cord  passing  over  the  pulley.  If  the  ^^. 
weight,   W,   be   greater   than   W\   the   former  Li^ 

will  descend,  and  draw  the  latter  up. 

In  this  case,  the  moving  force  is  the  difference  of  the 
weights,  W  and  Wj  the  mass  moved  is  the  sum  of  the 
masses  of  TF  and  W\  together  with  that  of  the  pulley  and 
connecting  cord.  The  different  parts  of  the  pulley  move 
with  different  velocities,  but  the  effect  of  its  mass  may  be 
replaced  by  that  of  some  other  mass  at  the  circumference 
of  the  pulley.  Denoting  this  mass,  together  with  the  mass 
of  the  cord,  by  r.i",  and  the  masses  of  W  and  W  by  m  and 
m',  we  have — to  represent  the  entire  mass  moved — the  ex- 
pression m  +  m'  +  wi",  and  for  the  moving  force  we  shall 
have  {m  —  m')g;  hence,  by  the  rule,  the  acceleration,  de- 
noted by  g',  is  equal  to, 

m  —  m' 
m  +  m  +  m  ^ 

This  force  being  constant,  the  motion  produced  by  it  is 
uniformly  varied,  and  the  circumstances  of  that  motion 
will  be  made  known  by  substituting  the  above  expression 
for/,  in  equation  (49)  and  (50). 

Atwood's  Machine. 

107.  Atwood's  machine  is  a  contrivance  to  illustrate 
the  laws  of  falling  bodies.     It  consists  of  a  vertical  post, 


128 


MECHANICS. 


AB,  about  12  feet  in  height,  supporting,  at 

ito  upper  extremity,  a  fixed   pulley,  A.      To 

obviate,  as  much  as  possible,  the  resistance  of 

friction,  the  axle  is  made  to  turn  on  friction 

rollers.     A  silk  string  passes  over  the  pulley, 

and  at  its  extremities  are  fastened  two  equal 

weights,  C  and  D.     In  order  to  impart  motion 

to  the  weights,  a  small  weight,  G,  in  the  form 

of  a  bar,  is  laid  on  C,  and  by  diminishing  its 

mass,  the    acceleration    may  be   rendered   as 

small  as  desirable.     The  rod,  AB,  graduated 

to  feet  and  decimals,  is  provided  with  sliding 

stages,  B  and  F;  the  upper  one  is  in  the  form 

of  a  ring,  which  will  permit  C  to  pass,  but  not 

G;  the  lower   one   is  in  the  form  of  a  plate,  which  is 

intended  to  intercept  the  weight  0.     Connected  with  the 

instrument  is  a  seconds  pendulum  for  measuring  time. 

Suppose  the  weights,  C  and  D,  each  equal  to  150  grains, 
the  weight  of  the  bar  24  grains,  and  let  a  weight  of  62 
grains,  placed  at  the  circumference  of  the  pulley,  produce 
the  same  resistance  by  its  inertia  as  that  actually  pro- 
duced by  the  pulley  and  cord.     Then  will  the  fraction 

?/?  —  7n' 


in-  +  m  +  m 
by  32J-,  gives  g 
and  (54),  gives. 


77  become  equal  to  ^\;   and  this,  multiplied 
2.     This  value,  substituted  for  g,  in  (53) 

V  =  2t,  and  h  =  T. 

If,  in  these  equations,  we  make  t  =  1  sec,  we  have  h  =  1, 
and  V  =  2.  If  we  make  t  =  2  sec,  we,  ni  like  manner, 
have  h  =  4,  and  v  =  4:.  If  we  make  t  =  3  sec,  we  have 
U  =  9,  and  v  =  Q,  and  so  on.  To  verify  these  results  ex- 
perimentally, commencing  with  the  first: — The  weight,  T, 
is  drawn  up  till  it  comes  opposite  the  0  of  the  graduated 


RECTILINEAR   AND    PERIODIC    3I0TI0X.  120 

scale,  and  the  bar,  G,  is  placed  on  it.  The  weight  thus  set 
is  held  in  its  place  by  a  spring.  The  ring,  E,  is  set  at  1 
foot,  and  the  stage,  F,  at  3  feet  from  the  0.  When  the  pen- 
dulum reaches  one  of  its  extreme  limits,  the  spring  is 
pressed  back,  the  weight,  CG,  descends,  and  as  the  pendu- 
lum completes  its  vibration,  the  bar,  6^,  strikes  the  ring,  and 
is  retained.  The  acceleration  then  becomes  0,  and  C  moves 
on  uniformly,  with  the  velocity  acquired,  in  the  first  sec- 
ond; and  it  will  be  observed  that  C  strikes  the  second 
stage  just  as  the  pendulum  completes  its  second  vibration. 
Had  F  been  set  at  5  feet  from  the  0,  C  would  have  reached 
it  at  the  end  of  the  third  vibration  of  the  pendulum.  Had 
it  been  7  feet  from  the  0,  it  would  have  reached  it  at  the 
end  of  the  fourth  vibration,  and  so  on. 

To  verify  the  next  result,  we  set  the  ring,  E,  at  four  feet 
from  the  0,  and  the  stage,  F,  at  8  feet  from  the  0,  and  pro- 
ceed as  before.  The  ring  Avill  intercept  the  bar  at  the  end 
of  the  second  vibration,  and  the  weight  will  strike  the 
stage  at  the  end  of  the  third  vibration,  and  so  on. 

By  making  the  weight  of  the  bar  less  than  24  grains,  the 
acceleration  is  diminished,  and,  consequently,  the  spaces 
and  velocities,  correspondingly  diminished.  The  results 
may  be  verified  as  before. 

Motion  of  Bodies  on  Inclined  Planes. 
108.  If  a  body  be  placed  on  an  inclined  plane,  and 
abandoned  to  the  action  of  its  own  weight,  it  will  either 
slide  or  roll  down  the  plane,  provided  there  be  no  friction 
between  it  and  the  plane.  If  the  body  is  spherical,  it 
will  roll,  and  in  this  case  friction  may  be  disregarded.  Let 
the  weight  of  the  body  be  resolved  into  two  components, 
one  perpendicular  to  the  plane,  and  the  other  parallel  to 
it:  the  plane  of  these  components  will  be  vertical,  and 


130  MECHANICS. 

also  perpendicular  to  the  given  plane.  The  effect  of  the 
first  component  will  be  counteracted  by  the  resistance  of 
the  plane,  whilst  the  second  will  act  as  a  constant  force, 
urging  the  body  down  the  plane.  The  force  being  con- 
stant, the  body  will  have  a  uniformly  varied  motion,  and 
equations  (53)  and  (54)  will  be  applicable.  The  accelera- 
tion may  be  found  by  projecting  the  acceleration  due  to 
gravity  on  the  inclined  plane. 

Let  AB  represent  the  inclined  plane,  and  P  the  centre 
of  gravity  of  a  body  resting  on  it.  Let  PQ  represent  the 
force  of  gravity,  denoted  by  g,  and 
PE,  its  component,  parallel  to   ^^,  _  f^ 

PS  being  the  normal  component. 

Denote  FE  by  g',  and  the  angle  3.  ^^'^ 
ABC  by  a.     Then,  since  FQ  is  per- 
pendicular  to  BCy  and  QE  to  AB,  the 
angle,  EQF,  is  equal  to  ABC,  or  to  a.     From  the  right- 
angled  triangle,  FQE,  we  have, 

g'  =  ^sina. 

But  the  triangle,  ABC,  is  right-angled,  and,  if  we  denote 
its  height,  AC,  by  h,  and  its  length,  AB,  by  I,  we  shall 

have  sina  =-,  which,  being  substituted  above,  gives, 

9'  =  '4 (63) 

This  value  of  g'  is  the  acceleration  due  to  the  moving 
force.  Substituting  it  for  /,  in  equations  (51)  and  (52), 
we  have, 

v  =  v'  +  Y^, 


RECTILINEAR   AND    PERIODIC    MOTION.  131 

If  the  body  start  from  rest  at  A,  taken  as  the  origin  of 
spaces,  then  will  v'  =  0,  and  s'  =  0,  giving, 

v  =  ^-ff (64) 

To  find  the  time  required  for  a  body  to  move  from  the 
top  to  the  bottom  of  the  plane,  make  s  =  I,  in  (65) ;  there 
will  result, 


'='P^  ••■'=VI (««) 


gh 

Hence,  the  time  varies  directly  as  the  letigth,  and  inversely 
as  the  square  root  of  the  height. 

For  planes  having  the  same  height,  but  different  lengths, 
the  radical  factor  of  the  value  of  t  remains  constant. 
Hence,  tlie  times  required  for  a  body  to  move  down  planes 
having  the  same  height,  are  to  each  other  as  their  lengths. 

To  determine  the  velocity  with  which  a  body  reaches 
the  bottom  of  the  plane,  substitute  for  t,  in  equation  (64), 
its  value  taken  from  equation  (66).  We  have,  after  re- 
duction, 

V  =  ^"itgh. 

But  this  is  the  velocity  due  to  the  height  h,  (Art.  104). 
Hence,  the  velocity  generated  in  a  body  whilst  movi^ig 
down  an  inclined  plane,  is  equal  to  that  generated  in  falling 
through  the  height  of  the  plane. 

Examples. 

1.  An  inclined  plane  is  10  feet  long  and  1  foot  high.  How  long 
will  it  take  for  a  body  to  move  from  top  to  bottom,  and  what  veloc- 
ity will  it  acquire  ? 


132  ,'  MECHANICS. 

SOLUTION. 

We  have,  from  equation  (66), 

V  gh' 
substituting  for  ^,  10,  and  for  7i,  1,  we  have, 

t  =  24  seconds,  nearly. 
From  the  formula,  v  =  V2gh,  we  have,  by  making  A  =  1, 

2.  How  far  will  a  body  descend  from  rest  in  4  seconds,  on  an  in- 
clined plane  whose  length  is  400  feet,  and  whose  height  is  300  feet  ? 

Ans.  193  ft. 

3.  How  long  will  it  take  a  body  to  descend  100  feet  on  a  plane 
whose  length  is  150  feet,  and  whose  height  is  60  ?         Ans.  3.9  sec. 

4.  There  is  a  track,  2i  miles  in  length,  whose  inclination  is  1  in  35. 
What  velocity  will  a  car  attain,  in  running  the  length  of  the  road,  by 
its  own  weight,  hurtful  resistances  being  neglected? 

Ans.  155.75  ft.,  or,  106.2  m.  per  hour. 

5.  A  railway  train,  having  a  velocity  of  45  miles  per  hour,  is  de- 
tached from  the  locomotive  on  an  ascending  grade  of  1  in  200.  How 
far,  and  for  what  time,  will  the  train  continue  to  ascend  the  inclined 
plane  ? 

SOLUTION. 

We  find  the  velocity  66  ft.  Hence,  66  =  V2gh  ;  or,  h  =  67.7  ft.  for 
the  vertical  height.  Hence,  67.7  X  200=  13,540  ft.,  or,  2.5644  m., 
the  distance  the  train  will  proceed.     We  have, 

t  =  l\/  —-  =  410.3  sec,  or,  6  min.  50.3  sec, 
\  gk 

the  time  required  to  come  to  rest. 

0.  A  body  weighing  5  lbs.  descends  vertically,  and  draws  a  weight 
of  6  lbs.  up  an  inclined  plane  of  45°.  How  far  will  the  first  body 
descend  in  10  seconds? 

SOLUTION. 

The  moving  force  is  5  lbs.  —6  lbs.  x  sin  45" ;  consequently  the 
acceleration  is,  (Art.  106), 

.-.  »  =  jyr  =  lll/(.,  nearly. 


RECTILINEAR   AND    PERIODIC    MOTIOX.  133 

Motion  of  a  Body  do^vn  a  succession  of  Inclined  Planes. 

1C9.  If  a  body  start  from  the  top  of  an  inclined  plane, 
with  an  initial  velocity,  v',  it  will  reach  the  bottom  with  a 
velocity  equal  to  the  initial  velocity,  plus  that  acquired 
whilst  on  the  plane.  This  velocity,  called  the  terminal 
velocity,  will  be  equal  to  that  which  the  body  would  acquire 
in  falling  through  the  height  due  to  the  initial  velocity, 
phis  the  height  of  the  plane.  Hence,  if  a  body  start  from 
rest  at  A,  and,  after  having  passed 
over  one  plane,  ^1J5,  enter  on  a  second, 
BC,  without  loss  of  velocity,  it  will 
reach  the  bottom  of  the  second  plane 
with  the  same  velocity  that  it  would  Fig.  94. 

have  acquired  by  falling  through  DC,  the  sum  of  the  heights 
of  the  two  planes.  Were  there  a  succession  of  inclined 
planes,  so  arranged  that  there  would  be  no  loss  of  velocity 
in  passing  from  one  to  another,  it  might  be  shown,  by 
similar  reasoning,  that  the  terminal  velocity  would  be  that 
due  to  the  vertical  distance  of  the  terminal  point  below  the 
point  of  starting. 

By  a  course  of  reasoning  analogous  to  that  employed  in 
discussing  the  motion  of  bodies  projected  vertically  up- 
ward, it  might  be  shown  that,  if  a  body  Avere  projected 
upward,  in  the  direction  of  the  lower  plane,  with  the  ter- 
minal velocity,  it  would  ascend  along  the  several  planes  to 
the  top  of  the  highest  one,  where  the  velocity  would  be  0. 
The  body  would  then,  under  the  action  of  its  own  weight, 
retrace  its  path  in  such  manner  that  the  velocity  at  every 
point  in  descending  w^ould  be  the  same  as  in  ascending,  but 
in  a  contrary  direction.  The  time  occupied  in  passing  over 
any  part  of  the  path  in  descending,  would  be  equal  to  that 
occupied  in  passing  over  the  same  portion  in  ascending. 


134-  MECHANICS. 

In  the  preceding  discussion,  we  have  supposed  that  there 
is  no  loss  of  velocity  in  passing  from  one  plane  to  another. 
To  ascertain  under  what  circumstances  this  condition  will 
be  fulfilled,  let  us  take  two  planes,  AB  and  BC.  Prolong 
CB  upward,  and  denote  the  angle,  ABU,  by  <p.  Denote 
the  velocity  of  the  body  on  reaching  B,  by  v'.  Let  v'  be 
resolved  into  two  components,  one  in  the  direction  of  BC, 
and  the  other  at  right  angles  to  it.  The  effect  of  the  latter 
is  destroyed  by  the  resistance  of  the  plane,  and  the  former 
is  the  effective  velocity  in  the  direction  of  the  plane,  BC. 
From  the  rule  for  resolution  of  velocities,  we  have,  the 
effective  component  of  v'  equal  to  v'  cos:p.  Hence,  the  loss 
of  velocity  is  v'—  v'  cos?,  or,  v'  (1  —  cos?)).  But  when  9  is 
infinitely  small,  its  cosine  is  1,  and  there  is  no  loss  of  velo- 
city. Hence,  the  loss  of  velocity  due  to  change  of  direction 
Avill  be  0,  when  the  path  is  a  curved  line.  The  principle  is 
general,  and  may  be  enunciated  as  follows : 

When  a  body  is  constrained  to  describe  a  curvilinear  path, 
there  is  no  loss  of  velocity  due  to  change  of  direction  of  the 
body's  motion. 

Periodic   Motion. 

110.  Periodic  motion,  is  a  species  of  variable  motion,  in 
which  the  spaces  described  in  certain  equal  periods,  are 
equal.  An  example  of  this  kind  of  motion  is  found  in 
curvilinear  vibration.  Let  ABC  be  a  vertical  curve,  sym- 
metrical with  respect  to  DB.  Let 
AC  he  horizontal,  and  denote  UB 
by  h.    If  a  body  were  placed  at  ^, 

and  abandoned  to  the  action  of  its    c^- 

own  weight,  being  constrained  to        cN;;^ 
remain  on  the  curve,  it  would,  in 
accordance  with  the  principles  of  ^ 


RECTILINEAK    AND    PERIODIC    MOTION.  135 

the  last  article,  move  toward  B  with  an  accelerated  mo- 
tion, and,  on  arriving  at  B,  would  possess  a  velocity  due 
to  the  height  h.  By  virtue  of  its  inertia,  it  would  ascend 
the  branch,  BC,  with  retarded  motion,  and  would  finally 
reach  C,  where  its  velocity  would  be  0.  The  body  would 
then  be  in  the  same  condition  that  it  was  at  A,  and 
would,  consequently,  descend  to  B,  and  again  ascend  to 
A,  whence  it  would  again  descend,  and  so  on.  Were 
there  no  retarding  causes,  the  motion  would  continue  for- 
ever. From  what  precedes,  it  follows  that  the  time  occupied 
by  the  body  in  passing  from  ^  to  ^  is  equal  to  that  in 
passing  from  B  to  C,  and  also  the  time  in  passing  from  C 
to  B  is  equal  to  that  in  passing  from  B  to  A.  Further, 
the  velocities  of  the  body  when  at  G  and  H,  two  points  on 
the  same  horizontal,  are  equal,  either  one,  being  that  due  to 
the  height  EK.  These  principles  are  used  in  discussing 
the  pendulum. 

Angular  Velocity,  and  Angular  Acceleration. 

111.  When  a  body  revolves  about  an  axis,  its  points 
being  at  different  distances  from  the  axis,  will  have  differ- 
ent velocities.  The  angular  velocity  is  the  velocity  of  a 
point  whose  distance  from  the  axis  is  1.  To  obtain  the 
velocity  of  any  other  point,  we  multiply  its  distance  from 
the  axis  by  the  angular  velocity.  The  force  of  gravity  acts 
uniformly  on  the  different  points  of  a  body,  and  the  m- 
pressed  acceleration  is  the  same  for  all  the  particles.  If  the 
body  is  constrained  to  turn  about  a  horizontal  axis,  the 
effective  acceleration  of  different  particles  will  depend  on 
their  distance  from  the  axis.  The  effective  acceleration  of 
a  point,  at  a  unit's  distance  from  the  axis,  is  called  the 
angular  acceleration  of  the  body. 


13G 


MECHANICS. 


The  Simple  Pendulum. 

112.  A  pendulum  is  a  heavy  body  suspended  from  a 
horizontal  axis,  about  which  it  is  free  to  vibrate. 

To  investigate  the  circumstances  of  vibration,  let  us  first 
consider  the  hypothetical  case  of  a  material  point,  vibrat- 
ing about  an  axis  to  which  it  is  attached  by  a  rod  destitute 
of  weight.  Such  a  pendulum  is  called  a  simple  ienlu- 
LU3I.  The  laws  of  vibration  in  this  case  will  be  identical 
with  those  explained  in  Art.  110,  the  arc,  ABC,  being  an 
arc  of  a  circle. 

Let  ABC  be   the  arc   through   which  vibration  takes 
place,  and  denote  its  radius,  DA,  by  I.     The  angle,  ADC, 
is  called  the  amplitude  of  vibration;  half  of  this  angle, 
ADB,  is  called  the  angle  of  devia- 
tion. 

If  the  point  start  from  rest  at  A, 
it  will,  on  reaching  any  point,  H, 
have  a  velocity  ii,  due  to  the  height, 
^Z,  denoted  by /i,  (Art.  104).  Hence, 

V  =  v^A (67) 

Let  the  angle  of  deviation  be  so 
small,  that  the  chords  of  the  arcs, 
AB  and  HB,  may  be  considered 
equal  to  the  arcs  themselves.  We 
shall  have  (Legendre,  Bk.  IV.,  Prop. 
XXIIL,  Oor.), 


Pig.  96. 


AB^  =  2lX  EB,  and  HB^  =  2Z  X  KB, 


whence,  by  subtraction, 

W  ~  ITS'  =  21{EB  -  KB)  =  2lX  h. 


RECTILINEAR   A]S"D    PERIODIC    MOTIOl^.  137 

Denoting  AB  \)j  a,  HB  by  x,  and  solving  the  last  equa- 
tion, we  have, 

Substituting  this  value  of  h,  in  (67),  it  becomes, 
v^\/^j{ce-x') (68) 

Now  let  us  develop  the  arc,  ABC,  into  a  straight  line, 
A'B'C,  and  suppose  a  point  to  start  from  A'  at  the  same 
time  that  the  pendulum  starts  from  yl,  and  to  vibrate  back 
and  forth  upon  A'B'C  with  the  same  velocities  as  the 
pandulnm  ;  then,  when  the  pendulum  is  at  any  point,  H, 
this  point  will  be  at  a  correspondhig  point,  H',  and  the 
times  of  vibration  of  the  two  will  be  the  same. 

To  find  tlie  time  of  vibration  along  A'B'C,  describe  on 
it  a  semi-circle,  A' 310',  and  suppose  a  third  point  to  start 
from  A '  at  tlie  same  time  as  the  second,  and  to  move  uni- 
formly around  the  arc  with  a  velocity  equal  to  ay  ^.  Then 

c 

will  the  time  required  for  this  particle  to  reach  C  be  equal 
to  the  space  divided  by  the  velocity,  (Art.  102).  Denoting 
this  time  by  t,  and  remembering  that  A'B'  =  a,  we  have, 


'  =  .57"4 


/ 


f/ 


I 

Make  IT'B'  =  x,  and  draw  H'M  perpendicular  to  A'C, 
and  at  M  decompose  the  velocity  of  the  third  particle,  MT, 
into  two  components,  MJV  and  MQ,  parallel  and  perpen- 
dicular to  A'C. 

We  have,  for  the  horizontal  component, 

MN  =  MTgo^  TMN (69) 


138  MECHANICS. 

But, MT  =ayy,  and  because MT and ifiV^are perpendic- 
ular to  ^'Jf  and  H'M,  we  have,  cos  TMN  —  cos  B'MH'  = 
-dttt'       But    B'M  —  a,   and   H'M  =  /^/^^  _  ^2 ;    hence, 

cos  TMN  = — .     Substituting  these  values  in  equa- 

ci 

tion  (69),  we  have,  for  the  horizontal  velocity, 


which  is  the  same  value  as  that  obtained  for  v,  in  equation 
(68).  Hence,  we  infer  that  the  horizontal  velocity  of  the 
third  point  is  always  equal  to  that  of  the  second  point, 
consequently  the  times  required  to  pass  from  A'  to  C  must 
be  equal ;  that  is,  the  time  of  vibration  of  the  second  point. 


A. 

noting  this  time  by  t,  we  have, 


and  consequently  of  the  pendulum,  must  be  it  y  —,    De- 


./^ 


t  =  'KV  - (70) 

Hence,  the  time  of  vibration  of  a  simple  pendulum  is 
equal  to  the  number  3.1416,  multiplied  into  the  square  root 
of  the  quotient  obtained  by  dividing  the  length  of  the  pen- 
dulum, by  the  force  of  gravity. 

For  a  pendulum,  whose  length  is  V,  we  shall  have, 

f  =  -^7j (71) 

From  equations  (70)  and  (71),  we  have,  by  division, 
7,  =  ^,;or,  ^^:  r  ::  vT:a/7' (72) 


I 


RECTILINEAR   AND    PERIODIC    MOTION.  139 

That  is,  the  twies  of  vibration  of  simple  pendulums,  are 
to  each  other  as  the  square  roots  of  their  lengths. 

If  we  suppose  the  lengths  of  two  pendulums  to  be  the 
same,  but  the  force  of  gravity  to  vary,  as  it  does  in  different 
latitudes,  and  at  different  elevations,  we  shall  have, 

t  —  It  \/  —,  and  t"  =  'j^ 

g 

Whence,  by  division, 
i.,=  * /f',     or,    if  :  if"  :  :  V7'  :  V7  ■  •  ■  (73) 

That  is,  the  times  of  vibration  of  the  same  pendulum,  at 
different  places,  are  to  each  other  inversely  as  the  square 
roots  of  the  forces  of  gravity  at  the  places. 

If  we  suppose  the  times  of  vibration  to  be  the  same,  and 
the  force  of  gravity  to  vary,  the  lengths  will  vary  also,  and 
we  shall  have, 

i5  =  cr|/I        and       t=.'.\/^,. 
9  9 

Equating  these  values,  and  squaring,  we  have, 

1  =  1;     or,     I  :  V  ::  g  :  q' (74) 

9      9 

That  is,  the  lengths  of  pendulums  that  vibrate  in  equal 
times  at  different  places,  are  to  each  other  as  the  forces  of 
gravity  at  those  places. 

Vibrations  of  equal  duration  are  called  isochronal. 

De  I'Ambert's  Principle.  » 

113.  When  several  bodies  are  rigidly  connected,  it  often 
happens  that  they  are  constrained  to  move  in  a  different 
manner  from  what  they  would,  if  free.  Some  move  faster 
and  some  slotver  than  they  would,  were  it  not  for  the  con- 


140  MECHANICS. 

nection.  In  the  former  case  there  is  a  gain,  and  in  the 
latter  a  loss,  of  moving  force,  in  consequence  of  the  connec- 
tion. It  is  obvious  that  the  resultant  of  all  the  impressed 
forces  is  equal  to  that  of  all  the  effective  forces,  for  if  the 
latter  were  reversed,  they  would  hold  the  former  in  equi- 
librium. Hence,  all  the  moving  forces  lost  and  gained  in 
consequence  of  the  connection  are  in  equilibrium. 
This  is  de  r Amber fs  principle. 

The    Compound   Pendulum. 

114.  A  compound  pendulum  is  a  body  free  to  vibrate 
about  a  horizontal  axis,  called  the  axis  of  suspension.  The 
straight  line  drawn  from  the  centre  of  gravity  of  the  pen- 
dulum perpendicular  to  the  axis  of  suspension  is  called 
the  axis  of  the  pendidum. 

In  practical  applications,  the  pendulum  is  so  shaped  that 
the  plane  through  the  axis  of  suspension  and  the  centre  of 
gravity  divides  it  symmetrically. 

Were  the  particles  of  the  pendulum  entirely  discon- 
nected, but  constrained  to  remain  at  invariable  distances 
from  the  axis  of  suspension,  we  should  have  a  collection 
of  simple  pendulums.  Those  at  equal  distances  from  the 
axis  would  vibrate  in  equal  times,  and  those  unequally  dis- 
tant would  vibrate  in  unequal  times.  The  particles  nearest 
the  jixis  would  vibrate  more  rapidly  than  the  compound 
pendulum,  and  those  most  remote  would  vibrate  slower; 
hence,  there  must  be  intermediate  points  that  would  vibrate 
in  the  same  time  as  the  pendulum.  These  points  lie  on  the 
Surface  of  a  circular  cylinder  whose  axis  is  that  of  suspen- 
sion ;  the  point  in  which  this  cylinder  cuts  the  axis  of  the 
pendulum  is  called  the  centre  of  oscillation.  If  the  entire 
mass  of  the  pendulum  wore  concentrated  at  this  point,  the 
time  of  its  vibration   would   be  unchanged.     Hence,  the 


RECTILINEAR   AND    PERIODIC    MOTION. 


141 


centre  of  oscillation  of  a  pendulum  is  a  point  of  its  axis,  at 
which,  if  the  mass  of  the  pendulum  were  concentrated,  its 
time  of  vibration  would  be  unchanged.  A  line  drawn 
through  this  point,  parallel  to  the  axis  of  suspension,  is 
called  the  axis  of  oscillation.  The  distance  between  the 
axis  of  oscillation  and  the  axis  of  suspension  is  the  length 
of  an  equivalent  simple  pendulum — that  is,  of  a  simple  pen- 
dulum, whose  time  of  vibration  is  the  same  as  that  of  the 
compound  pendulum. 


Angular  Acceleration  of  a  Compound  Pendulum. 

115.  Let  CK  be  a  compound  pendulum,  C  its  axis  of 
suspension,  G  its  centre  of  gravity,  and  suppose  the  plane 
of  the  paper  to  pass  through  the  centre  of  gravity,  G,  and 
perpendicular  to  the  axis,  C.  We  may  regard  the  pendulum 
as  made  up  of  infinitely  small  filaments,  parallel  to  the 
axis  of  suspension,  and  consequently  perpendicular  to  the 
paper.  The  circumstances  of  vibration  will  be  unchanged 
if  we  suppose  each  element  to  be  concentrated  in  the  point 
where  it  meets  the  plane  of  the  paper.  Denote  the  mass 
of  any  element,  as  S,  by  in,  its  distance  from  C,  by  r,  and  the 
mass  of  the  pendulum  by  M. 

Through  0  draw  a  horizontal  line,  CB,  and  draw  SH, 
GA,  and  OB  perpendicular  to  it. 
On  HS  prolonged,  take  SJ^  to  rep- 
resent the  moving  force  impressed 
on  S.  Then  will  8E  be  equal  to  mg, 
(Art.  18),  and  its  moment  with  re- 
spect to  0  will  be  mg  X  HC.  Denote 
the  angular  acceleration  by  oo  ;  then 
will  the  actual  acceleration  of  >S', 
in  the  direction  perpendicular  to 
^'6',  bo  e<iual  to  rcj,  and  the  effective 


Fig.  97. 


142  MECHAKICS. 

moving  force  to  mroo  ;  because  this  force  acts  at  right  angles 
to  SO,  its  moment  is  equal  to  7nr^oo.  Because  7ng  is  the 
moving  force  impressed  on  S,  and  mroo  the  effective  moving 
force,  the  expression,  mg  —  mroo,  will  be  the  moving  force 
lost  or  gained  by  S  in  consequence  of  its  connection  with 
the  other  particles.  There  will  be  a  loss  when  mg  is  greater 
than  mroo,  and  a  gain  when  mg  is  less  than  fnrao.  The 
moment  of  this  force  with  respect  to  C  is  equal  to 
mg  X  CH  —  mr^oo.  Similar  expressions  may  be  found  for 
each  of  the  elementary  particles  of  the  pendulum. 

By  de  I'Ambert's  principle,  the  moving  forces  lost  and 
gained,  in  consequence  of  the  connection  of  the  parts,  are 
in  equilibrium ;  hence,  the  algebraic  sum  of  their  moments 
with  respect  to  an  axis,  C,  is  equal  to  0 — that  is, 

I{mg  X  CH)  -  I{m7-'oo)  =  0. 

But  GO  and  g  are  the  same  for  each  particle ;  hence, 

I{m  X  CH) 
-9- 


l\7nr') 
From  the  principle  of  moments,  we  have, 

I{m  X  CH)  =31  X  CA. 
Substituting  above,  we  have,  finally, 

MX  CA  .^„. 

-  =  ^(m-n-^ ^^'^ 

That  is,  the  a^igular  acceleration  varies  as,  CA,  the  lever 
arm  of  the  weight  of  the  pendulum. 

The  expression  I{mr'')  is  called  the  moment  of  inertia  of 
the  body  with  respect  to  the  axis  of  suspension,  Mg  is  the 
weight  of  the  body,  and  Mg  X  CA  is  the  moment  of  the 
weight,  with  respect  to  the  same  axis. 

Hence,  the  angular  acceleration  is  equal  to  the  moment 


RECTILINEAR    AND    PERIODIC    MOTION.  143 

of  the  iveigJit^  divided  by  the  moment  of  iJiertia,  both  loith 
respect  to  the  axis  of  susj)e7ision. 

Length  of  an  Equivalent  Simple  Pendulum. 

116.  To  find  the  length  of  a  simple  pendulum  that  will 
vibrate  in  the  same  time  as  the  given  compound  pendulum, 
let  0  be  the  centre  of  oscillation,  and  draw  OB  perpen- 
dicular to  CB.  Denote  CO  by  I,  and  CG  by  k.  Were  the 
entire  mass  concentrated  at  0,  we  should  have,  for  its 
moment  of  inertia,  Ml^,  for  the'  moment  of  the  mass, 
M  X  GB,  and  for  the  angular  acceleration, 

MxGB 

But  the  pendulum  is  to  vibrate  in  the  same  time,  whether 
it  exist  as  a  compound  pendulum,  or  as  a  simple  pendulum, 
its  mass  being  concentrated  at  its  centre  of  oscillation ;  the 
value  of  w  must,  therefore,  be  the  same  in  both  cases. 
Placing  the  value  just  deduced  equal  to  that  in  equation 
(75),  we  have, 

MxGB    _MxGA 

Mr     ^~  l\7nr')   ^'' 

whence,  by  reduction. 

From  the  similar  triangles,  GGA  and  GOB,  we  have, 

GB  _  I 

GA  ~  k' 

Substituting,  and  reducing,  we  have, 

^-^    Mk     (^^^ 


144 


MECHANICS. 


Reciprocity  of  Axes  of  Suspension  and  Oscillation. 

117.  Let  C  be  the  axis  of  suspension,  0  the  centre  of 
oscillation,  and  let  a  line  be  drawn  through  0  parallel  to 
the  axis  of  suspension.  This  line  is 
called  the  axis  of  oscillation.  Let 
the  plane  of  the  paper  be  taken  as 
before,  and  suppose  the  elements 
projected  on  it,  as  in  the  last  article. 

Let  S  be  any  element,  and  denote 
its  distance  from  the  axis  of  suspen- 
sion by  r,  and  from  the  axis  of  os- 
cillation by  t ;  denote  OC  by  /,  and 
the  angle  0  CS  by  (p. 

If  the  axis  of  oscillation  be  taken  as  an  axis  of  suspen- 
sion, and  the  length  of  the  corresponding  simple  pendulum 
denoted  by  I',  we  have,  from  the  preceding  article, 

I(mf) 


Fiff.  98. 


V 


(77) 


M{l-k) 

In  the  triangle,  OSC,  we  have, 

f  =  r*  +  r  -  2Wcos(p ; 

hence, 

I{mf)  =  I(mr')  +  I{mP)  -  2Z(mrcos(p)l 

But,  from  equation  (76),  we  have, 

I{mr')  =  MM; 

and  because  I  is  invariable,  we  have, 

^ml")  =  I{m)r  =  m; 

if  we  suppose  CO  horizontal,  rcos?,  the  projection  of  r 
on  CO,  will  be  the  lever  arm  of  m,  and  the  expression, 
J(w2rcos^),  will  be  the  algebraic  sum  of  the  moments  of 


RECTILINEAR   AND    PERIODIC    MOTION.  145 

the  elementary  masses  with  respect  to  C ;  hence,  we  shall 

have, 

2'(wrcos(p)^  =  Mkl, 

Substituting  for  these  expressions  their  values  above,  and 
putting  the  value  of  ^{mf),  thus  found,  in  (77),  we  have, 

^  Mkl  +  Mr-  2Mkl  _  M{r-kl)^ 
M\l-k)  ~    M(l-ky 

or, 

l'  =  l 

Hence,  the  axes  of  susjyension  and  oscillation  are  con- 
vertible ;  that  is,  if  either  be  taken  as  an  axis  of  suspension, 
the  other  ivill  he  the  axis  of  oscillation. 

This  property  of  the  compound  pendulum  is  employed 
to  determine  the  length  of  the  seconds'  pendulum,  and  the 
value  of  the  force  of  gravity  at  different  places  on  the  sur- 
face of  the  earth. 

A  straight  bar  of  iron,  CD,  is  provided  with  knife-edge 
axes,  A  and  B,  of  hardened  steel,  at  right  angles  to  the  axis 
of  the  bar,  and  having  their  edges  turned  toward  c 
each  other.  These  axes  are  so  placed  that  they  are 
not  symmetrical  with  respect  to  the  bar.  The 
pendulum  thus  constructed  is  suspended  on  hori- 
zontal plates  of  polished  agate,  and  allowed  to 
vibrate  about  each  axis  in  turn  till,  by  filing  away 
one  end  of  the  bar,  the  times  of  vibration  about  the 
axes  are  made  equal.  The  distance,  ^^,  is  then 
the  length  of  a  simple  pendulum  that  will  vibrate  ^ig-  ^■ 
in  the  same  time  as  the  bar,  about  either  axis.  The  adjust- 
ment may  also  be  made  by  using  a  sliding  piece,  that  can 
be  made  fast  to  the  bar,  by  a  clamp-screw. 

To  employ  the  pendulum  thus  adjusted  to  find  the  length 
of  a  seconds'  pendulum  at  any  place,  the  pendulum  is  sus- 
pended, and  allowed  to  vibrate  through  a  small  angle,  the 


146  MECHANICS. 

number  of  vibrations  counted,  and  the  time  noted  by  a 
chronometer.  The  time  divided  by  the  number  of  vibra- 
tions, gives  the  time  of  a  single  vibration.  The  distance 
between  the  axes,  measured  by  a  scale  of  equal  parts,  gives 
the  length  of  the  corresponding  simple  pendulum.  To  find 
the  length  of  the  simple  seconds'  pendulum,  we  make  use 
of  proportion  (72),  substituting  in  it  for  t'  and  V  the  values 
just  found,  and  for  t,  1  second ;  the  remaining  quantity  /, 
may  be  found  by  solving  the  proportion.  This  value  is  the 
length  of  the  seconds'  pendulum  at  the  place  where  the 
observation  is  made.  In  making  the  observations,  a  variety 
of  precautions  must  be  taken,  and  several  corrections  ap- 
plied, the  explanation  of  which  does  not  fall  within  the 
scope  of  this  treatise.  By  a  series  of  carefully  conducted 
experiments,  it  was  found  that  the  length  of  a  seconds' 
pendulum  in  the  Tower  of  London  is  3.2616  ft,  or  39.13921 
in.  By  a  similar  course  of  proceeding,  the  length  of  the 
seconds'  pendulum  has  been  determined  for  a  great  number 
of  places  on  the  earth's  surface,  at  different  latitudes,  and 
from  these  the  corresponding  values  of  the  force  of  gravity 
at  those  points  have  been  determined,  according  to  the  fol- 
lowing principle : 

From  the  equation,  ^  =  -ry -,  we  find,  by  solving  with 

respect  to  g,  and  making  ^  =  1, 
g  =  it-l 

From  this  equation  the  value  of  g  may  be  found  at 
different  places,  by  substituting  for  I  the  length  of  the 
seconds'  pendulum  at  those  places.  By  comparing  the 
values  of  g,  it  is  found  that  they  are  everywhere  the  same 
on  the  same  parallel  of  latitude,  and  that  they  vary  in  pass- 
ing from  latitude  to  latitude. 


RECTILINEAR   AND    PERIODIC    MOTION.  147 

The  following  formula  for  determining  the  value  of  ^,  at 
different  places,  is  given  by  Prof.  Airy.  In  it  G  represents 
the  value  of  gravity  at  the  equator,  g  its  value  in  any  lati- 
tude, I. 

g=  G(l  +  .005133  sin^O (78) 

The  value  of  G  is  32.088  ft. ;  this  gives  for  gravity  at  the 
pole,  32.2527  ft. 

Practical  Application  of  the  Pendulum. 

118.  One  of  the  most  important  uses  of  the  pendulum  is 
to  regulate  the  motion  of  clocks.  A  clock  consists  of  a 
train  of  wheelwork,  the  last  wheel  of  the  train  connecting 
with  a  pendulum-rod  by  a  piece  of  mechanism  called,  an 
escapement.  The  wheelwork  is  kept  in  motion  by  a  descend- 
ing weight,  or  by  the  elastic  force  of  a  spring,  and  the  wheels 
are  so  arranged  that  one  tooth  of  the  last  wheel  in  the  train 
escapes  from  the  pendulum-rod  at  each  vibration  of  the 
pendulum,  or  at  each  heat.  The  number  of  beats  is  ren- 
dered visible  on  a  dial-plate  by  indices,  called  hands. 

On  account  of  expansion  and  contraction,  the  length  of 
the  pendulum  is  liable  to  variation,  which  gives  rise  to 
irregularity  in  the  times  of  vibration.  To  obviate  this,  and 
to  render  the  times  of  vibration  uniform,  several  devices 
have  been  resorted  to,  giving  rise  to  what  are  called  com- 
pensatiyig  pendulums.  We  shall  indicate  two  of  the  most 
important  of  these,  observing  that  the  remaining  ones  are 
nearly  the  same  in  principle,  differing  only  in  mode  of 
application. 

Graham's  Mercurial  Pendulum. 

119.  Graham's  mercurial  pendulum  consists  of  a  rod  of 
steel  about  42  inches  long,  branched  toward  its  lower  end, 
to  embrace  a  cylindrical  glass  vessel  7  or  8  inches  deep. 


148  MECHANICS. 

and  having  between  6  and  7  inches  of  this  depth  filled  with 
mercury.  The  exact  quantity  of  mercury,  being  dependent 
on  the  weight  and  expansibility  of  the  other  parts  of 
the  pendulum,  may  be  determined  by  experiment  in  each 
case.  When  the  temperature  increases,  the  steel  rod  is 
lengthened,  and,  at  the  same  time,  the  mercury  rises  in  the 
cylinder.  When  the  temperature  decreases,  the  steel  bar 
is  shortened,  and  the  mercury  falls  in  the  cylinder.  By  a 
proper  adjustment  of  the  quantity  of  mercury,  the  effect 
of  the  lengthening,  or  shortening  of  the  rod  is  exactly 
counterbalanced  by  the  rising  or  falling  of  the  centre  of 
gravity  of  the  mercury,  and  the  axis  of  oscillation  is  kept 
at  an  invariable  distance  from  the  axis  of  suspension. 


Harrison's  Gridiron  Pendulum. 

120.  Harrison's  gridiron  pendulum  consists  of  five  rods 
of  steel  and  four  of  brass,  placed  alternately  with  each 
other,  the  middle  rod,  or  that  from  which  the 
doh  is  suspended,  being  of  steel.  These  rods  are 
connected  by  cross-pieces  in  such  a  manner  that, 
whilst  the  expansion  of  the  steel  rods  tends  to 
elongate  the  pendulum,  or  lower  the  bob,  the 
expansion  of  the  brass  rods  tends  to  shorten  the 
pendulum,  or  raise  the  bob.  By  duly  propor- 
tioning the  sizes  and  lengths  of  the  bars,  the 
axis  of  oscillation  may  be  maintained  at  an  in- 
variable distance  from  the  axis  of  suspension. 
From  what  has  preceded,  it  follows  that  when-  Pig."  lOO. 
ever  the  distance  from  the  axis  of  oscillation  to  the  axis  of 
suspension  remains  invariable,  the  times  of  vibration  must 
be  absolutely  equal  at  the  same  place.  The  pendulums 
just  described  are  principally  used  for  astronomical  clocks. 


m 


Q 


KECTILINEAK    AKD    PERIOJJIC    MOTIOX.  149 

where  great  accuracy  and  uniformity  in  the  measure  of 
time  are  indispensable. 

Basis  of  a  System  of  Weights  and  Measures. 

121.  The  pendulum  is  of  further  importance,  in  a  prac- 
tical point  of  view,  in  furnishing  the  standard  that  has 
been  made  the  basis  of  the  English  system  of  weights  and 
measures. 

It  was  enacted  by  Parliament,  in  1824,  that  the  distance 
between  the  centres  of  two  gold  studs  in  a  certain  described 
brass  bar,  the  bar  being  at  a  temperature  of  62°  F.,  should 
be  an  "  imperial  standard  yard."  To  be  able  to  restore  it 
in  case  of  its  destruction,  it  was  enacted  that  the  yard 
should  be  considered  as  bearing  to  the  length  of  the  seconds' 
pendulum  in  the  latitude  of  London,  in  vacuum,  and  at 
the  level  of  the  sea,  the  ratio  of  3G  to  39.1393.  From  the 
yard,  every  other  unit  of  linear  measure  may  be  derived, 
and  thence  all  measures  of  area  and  volume. 

It  was  also  enacted  that  a  certain  described  brass  weight, 
made  in  1758,  and  called  2  lbs.  Troy,  should  be  regarded 
as  authentic,  and  that  a  weight  equal  to  one-half  that 
should  be  "the  imperial  standard  Troy  j^ound."  The 
^^jig^th  part  of  the  Troy  pound  was  called  a  gram,  of 
which  7000  constituted  a  pound  avoirdupois.  To  provide 
for  the  contingency  of  a  loss  of  the  standard,  it  was  con- 
nected with  the  system  of  measures,  by  enacting,  that  if 
lost,  it  should  be  restored  by  allowing  252.724  grains  for 
the  weight  of  a  cubic  inch  of  distilled  water,  at  62°  F.,  the 
water  being  weighed  in  vacuum  and  by  brass  weights. 
From  the  grain  thus  established,  all  other  units  of  weight 
may  be  derived. 

Our  own  system  of  weights  and  measures  is  the  same  as 
the  English. 


150  mechanics. 

Examples. 

1.  The  length  of  a  seconds'  pendulum  is  89.13921  in.  If  it  be 
shortened  0.130464  in.,  how  many  vibrations  will  be  gained  in  a  day 
of  24  hours? 

SOLUTION. 

The  times  of  vibration  of  pendulums  at  the  same  place,  are  as  the 
square  roots  of  their  lengths.  Hence,  the  number  of  vibrations  in 
any  given  time,  are  inversely  as  the  square  roots  of  their  lengths. 
If,  therefore,  we  denote  the  number  of  vibrations  gained  in  24  hours, 
or  86400  seconds,  by  x,  we  have, 

86400  :  86400  -\- x  :  :  \/39.008747  :  \/39. 13921. 
Whence,  x  =  144,  nearly.  Ans. 

2.  A  seconds'  pendulum  on  being  carried  to  the  top  of  a  mountain, 
was  observed  to  lose  5  vibrations  per  day  of  86400  seconds.  Re- 
quired the  height  of  the  mountain,  reckoning  the  radius  of  the  earth 
at  4000  miles. 

SOLUTION. 

The  squares  of  the  times  of  vibration,  at  two  points,  are  inversely 
as  the  forces  of  gravity  at  those  points.  But  the  forces  of  gravity  at 
the  points  are  inversely  as  the  squares  of  their  distances  from  the 
centre  of  the  earth.  Hence,  the  times  of  vibration  are  proportional 
to  the  distances  of  the  points  from  the  centre  of  the  earth ;  and,  con- 
sequently, the  number  of  vibrations  in  any  given  time,  as  24  hours, 
for  example,  will  be  inversely  as  those  distances.  If,  therefore,  we 
denote  the  height  of  the  mountain  in  miles  by  x,  we  have, 
86400  :  86405  :  :  4000  :  4000  -f  x. 

Whence,       x  =  1%^^  =  0.2315  miles,  or,  1222  feet.  Ans. 

3.  What  is  the  time  of  vibration  of  a  pendulum  whose  length  is  60 
inches,  when  the  force  of  gravity  is  32^  ft.  ?  Ans.  1.2387  sec. 

4.  How  many  vibrations  will  a  pendulum  36  inches  in  length  make 
in  one  minute,  the  force  of  gravity  being  the  same  as  before  V 

Ans.  62.53. 

5.  A  pendulum  makes  43170  vibrations  in  12  hours.  How  much 
must  it  be  shortened  that  it  may  beat  seconds  ? 

SOLUTION. 

We  shall  have,  as  in  example  1st, 


43170  :  43200  :  :  \/39.13921  :  v/39.13921  +  x. 
Whence,  x  =  0.0544  m,  Ans. 


RECTILIl^EAR   AND    PERIODIC   MOTION.  151 

G.  In  a  certain  latitude,  the  length  of  a  pendulum  vibrating  seconds 
is  39  inches.  What  is  the  length  of  a  pendulum  vibrating  seconds, 
in  the  same  latitude,  at  the  height  of  21000  feet  above  the  first  sta- 
tion, the  radius  of  the  earth  being  3960  miles?         Ans.  38.9218  in. 

7.  If  a  pendulum  make  40000  vibrations  in  6  hours,  at  the  level  of 
the  sea,  how  many  vibrations  will  it  make  in  the  same  time,  at  an 
elevation  of  10560  feet,  the  radius  of  the  earth  being  3960  miles  ? 

Ans.  39979.8. 

Centre  of  Percussion. 

122.  The  centre  of  percussion  of  a  suspended  body,  is 
the  point  at  which  an  impulse  may  be  applied,  perpen- 
dicular to  the  plane  through  it  and  the  axis,  without  shock 
to  the  axis.  This  point  is  identical  with  the  centre  of  oscil- 
lation. For,  suppose,  whilst  the  body  is  vibrating  about 
the  axis,  an  impulse  to  be  applied  at  the  centre  of  oscilla- 
tion, capable  of  generating  a  quantity  of  motion  equal  and 
directly  opposed  to  the  resultant  of  the  quantities  of  motion 
of  all  the  particles ;  the  direction  of  this  impulse  will  be 
opposite  to  the  motion  of  the  centre  of  oscillation,  that 
is,  perpendicular  to  the  plane  through  it  and  the  axis,  and 
it  is  obvious,  from  the  property  of  the  centre  of  oscillation, 
that  it  will  bring  the  body  to  rest  without  shock  to  the 
axis.  Were  the  same  impulse  applied  to  the  body,  at  rest, 
it  would  generate  a  quantity  of  motion  equal  to  that  de- 
stroyed, but  in  a  contrary  direction,  and  without  shock  on 
the  axis.  The  direction  of  the  impulse  remaining  the  same, 
no  matter  what  may  be  its  intensity,  there  will  be  no  shock. 
It  is  a  matter  of  common  observation,  that  if  a  rod  held 
in  the  hand  be  struck  at  a  certain  point,  the  hand  will  not 
feel  the  blow,  but  if  struck  at  any  other  point,  a  shock  will 
be  felt,  the  intensity  of  which  depends  on  the  intensity  of 
the  blow,  and  on  its  point  of  application. 


152  MECHANICS. 

Moment  of  Inertia. 

123.  The  moment  of  inertia  of  a  body  with  respect  to 
an  axis,  is  the  algebraic  sum  of  the  products  obtained  by 
multiplying  the  mass  of  each  elementary  particle  by  the 
square  of  its  distance  from  the  axis.  Denoting  the  moment 
of  inertia  with  respect  to  any  axis,  by  K,  the  mass  of  anj 
element  of  the  body,  by  m,  and  its  distance  from  the  axis^ 
by  r,  we  have,  from  the  definition, 

K  =  I{mr') (79) 

The  moment  of  inertia  varies,  in  the  same  body,  accord- 
ing to  the  position  of  the  axis.     To  investigate  the  law 
of  variation,  let  AB  represent  a  section  of 
the  body  by  a  plane  perpendicular  to  the 
axis ;  C,  the  point  in  which  this  plane  cuts 
the    axis;  and    G,  the  point  in  which   it 


cuts  a  parallel  axis  through  the  centre  of         Fig.  loi. 
gravity.     Let    P  be    any  element    of   the 
body,  whose  mass  is  m,  and  denote  PC  by  r,  PG  by  s, 
and  CG  by  k. 

From  the  triangle  CPG,  according  to  a  principle  of  trig- 
onometry, we  have, 

r'  =  s'  +  F  -  2skcosCGP. 
Substituting,  in  (79),  and  separating  the  terms,  we  have, 
K  =  I{ms')  +  2'(mF)  -  2I(7nskcosCGP). 

Or,  since  k  is  constant,  and  w(m)  =  if,  the  mass  of  the 
body,  we  have, 

K  =  I(7ns')  +  Mk'  -  2kI(msGosCGP), 

But  scos  CGP  =  GH,  the  lever  arm  of  the  mass  in,  with 
respect  to  the  axis  through  the  centre  of  gravity.  Hence, 
I(m.ficosCGP),  is  the  algebraic  sum  of  the  moments  of  all 
the  particles  of  the  body  with  respect  to  the  axis  through 


kectili:n^ear  akd  perioj)ic  motiox.  153 

the  centre  of  gravity;  but  from  the  principle  of  moments, 
this  is  0.     Hence, 

K=  I{ms^)  +  M¥ (80) 

The  first  term  of  the  second  member  is  the  moment 
of  inertia,  with  respect  to  the  axis  through  the  centre  of 
gravity. 

Hence,  the  moment  of  iiiertia  of  a  hody  with  respect  to 
any  axis,  is  equal  to  the  moment  of  inertia  with  respect  to 
a  parallel  axis  through  the  centre  of  gravity,  plus  the  mass 
of  the  body  into  the  square  of  the  distance  between  the  two 
axes. 

The  moment  of  inertia  is  least  possible  when  the  axis 
passes  through  the  centre  of  gravity.  If  any  number  of 
parallel  axes  be  taken  at  equal  distances  from  the  centre 
of  gravity,  the  moment  of  inertia  with  respect  to  each,  will 
be  the  same. 

The  moment  of  inertia  with  respect  to  any  axis,  may  be 
determined  experimentally  as  follows.  Make  the  axis 
horizontal,  and  allow  the  body  to  vibrate  about  it,  as  a  com- 
pound pendulum.  Find  the  time  of  a  single  vibration,  and 
denote  it  by  t.  This  value  of  t,  in  equation  (70),  makes 
known  the  value  of  I  Determine  the  centre  of  gravity, 
and  denote  its  distance  from  the  axis,  by  k.  Find  the 
mass  of  the  body,  and  denote  by  M. 

We  have,  from  equation  (76), 

MM  =  I{mr')  =  K. 

Substitute  for  M,  I,  and  h,  the  values  already  found,  and 
the  value  of  K  will  be  the  moment  of  inertia,  with  respect 
to  the  assumed  axis.  Subtract  from  this  the  value  of  Jf/Fr 
and  the  remainder  will  be  the  moment  of  inertia  with 
respect  to  a  parallel  axis  through  the  centre  of  gravity. 
The   moment   of  inertia   of   a   homogeneous    body   of 


154  MECHANICS. 

regular  figure,  is  most  readily  found  by  means  of  the  cal- 
culus. 

The  results  thus  determined,  in  a  few  of  the  more  com- 
mon cases  of  practical  mechanics,  are  appended. 

1.  The  moment  of  inertia  of  a  rod,  or  bar,  of  uniform 
thickness,  with  respect  to  an  axis  perpendicular  to  the 
length  of  the  rod,  is  given  by  the  formula, 

A^=--M(^j  +  d'\ (81) 

in  which,  K  is  the  moment  of  inertia,  M  the  mass  of  the 
rod,  21  its  length,  and  d  the  distance  of  the  centre  of 
gravity  from  the  axis. 

2.  The  moment  of  inertia  of  a  thin  circular  plate  about 
a  line  in  its  own  plane,  is  given  by  the  formula, 


=  M(^j  +  d'\ (82) 


in  which,  II,  M,  and  d,  are  the  same  as  before,  and  r  is  the 
radius  of  the  circular  plate. 

3.  The  moment  of  inertia  of  a  circular  plate,  with  refer- 
ence to  an  axis  perpendicular  to  its  plane,  is  given  by  the 
formula, 

K=M(^-hd'\ (83) 

in  which,  the  quantities  are  the  same  as  before. 

4.  The  moment  of  inertia  of  a  circular  ring,  with  refer- 
ence to  an  axis  perpendicular  to  its  plane,  is  given  by  the 
formula, 

K=m(^'^—  +d'\ (84) 

in  which,  r  and  r'  are  the  exterior  and  interior  radii  of  the 
ring. 


KEC^TI LINEAR   AND    PERIODIC    MOTION.  155 

5.  The  inomenfc  of  inertia  of  a  right  cylinder  with  re- 
spect to  an  axis  perpendicular  to  the  axis  of  the  cylinder, 
is  given  by  the  formula, 

A'=.l/(J  +  ^  +  ^) (85) 

in  which,  r  is  the  radius  of  the  base,  and  "M  the  length  of 
the  cylinder. 

By  making  d=  o  in  any  of  the  above  formulas,  we  find 
the  corresponding  moment  of  inertia  for  a  parallel  axis 
through  the  centre  of  gravity. 

Centre  and  Radius  of  Gyration. 

124.  The  centre  of  gyration  with  respect  to  an  axis,  is  a 
point  at  which,  if  the  entire  mass  of  a  body  be  concen- 
trated, its  moment  of  inertia  will  remain  unchanged.  The 
distance  of  this  point  from  the  axis  is  the  radius  of  gyra- 
tion. 

Let  M  denote  the  mass  of  a  body,  and  k'  its  radius  Of 
gyration ;  then  will  the  moment  of  inertia  of  the  concen- 
trated mass  with  respect  to  the  axis,  be  equal  to  Mk'"^ ;  but 
this  must,  by  definition,  be  equal  to  the  moment  of  inertia 
with  respect  to  the  same  axis,  or  ^{mr"^)',  hence. 


Jf^"  ^  I{r)ir^) ;  or,  k'  =  V^^^^ (86) 

That  is,  the  radius  of  gyration  is  eqioal  to  the  square  root 
of  the  quotient  obtained  hy  dividing  the  motnent  of  i?iertia 
by  the  mass. 

Since  M  is  constant  for  the  same  body,  the  radius  of 
gyration  will  be  least  possible  when  the  moment  of  inertia 
is  least  possible,  that  is,  when  the  axis  passes  through  the 
centre  of  gravity.  This  minimum  radius  is  called  the 
p?Hn('ipal  radius  of  gyration.     If  we  denote  the  principal 


156  MECHANICS. 

radius  of  gyration  by  k,  we  sliall  have,  from  the  examples 
of  article  (123),  the  following  results : 


Example  l...k'  =  T  ^  +  ^/';  h  =  iVi. 

Example  2 . . .  ^•'  =  t/  -  ^  d': 
4 


.=r 


Example  d...k'=  y^-  +  (f;  k  =  rV\. 


Example  4.  .  .^•'  =  |/  ^'"  ^  ^''  +  ^:         X'  =  |/^'  "^  ^" 


^•' 

.v'-^ 

2 

+  ^; 

y 

-^i 

4 

+  <f; 

Example  5 .  . .  ^•'  =  y  ^'  +i-  +  ^' ;  ^-  =  '/ ^ 

4        o  4 


+  -3- 


To  find  the  relation  between  the  length  of  an  equivalent 
simple  pendulum  and  the  principal  radius  of  gyration  of  a 
suspended  body,  let  us  replace  the  expression  J(mr''),  in 
equation  (76),  by  its  value  Mk'^  and  reduce.     We  fiad, 

1  =  ^       .'.      k'  =  Vkf; 

that  is,  the  centre  of  gravity,  the  centre  of  oscillation,  and 
the  centre  of  gyration,  are  on  a  common  perpendicular  to 
the  axis  of  suspension,  and  so  situated  that  the  distance  of 
the  last  from  an  axis  is  a  mean  proportional  between  the 
distances  of  the  other  two  from  the  same  axis. 


CHAPTER  VI. 


CURVILINEAR    AND    ROTARY    MOTION. 


Motion  of  Projectiles. 

125.  If  a  body  be  projected  obliquely  upward  in  a 
vacuum,  and  then  abandoned  to  the  force  of  gravity,  it 
will  be  continually  deflected  from  a  rectilinear  path,  and, 
after  describing  a  curvilinear  trajectory,  will  finally  reach 
the  horizontal  plane  from  which  it  started. 

The  starting-point  is  the  ;^oi?^^  of  projection  ;  the  dis- 
tance from  the  point  of  projection  to  the  point  at  which 
the  projectile  again  reaches  the  same  horizontal  plane  is 
the  range,  and  the  time  occupied  is  the  time  of  flight.  The 
only  forces  to  be  considered,  are 
the  initial  impulse  and  the  force 
of  gravity.  Hence,  the  trajec- 
tory will  lie  in  a  vertical  plane 
through  the  direction  of  the 
initial  impulse.  Let  CAB  be 
this  plane,  A  the  point  of  pro- 
jection, AB  the  range,  and  AC  o,  vertical  through  A. 
Take  AB  and  AC  as  co-ordinate  axes;  denote  the  angle 
of  projection,  DAB.  by  «,  and  the  velocity  due  to  the 
initial  impulse  by  v.  Resolve  v  into  two  components, 
one  in  the  direction  AC,  and  the  other  in  tlie  direction 
AB.  We  have,  for  the  former,  v  sina,  and,  for  the  latter. 
V  cosa. 

The  velocities,  and.  consequently,  the  spaces  described 


Fig.  102. 


158  MECHANICS. 

in  the  direction  of  the  co-ordinate  axes,  will  (Art.  12)  be 
entirely  independent  of  each  other.  Denote  the  space 
described  in  the  direction  A  C,  in  any  time  t,  by  y.  The 
circumstances  of  motion  in  this  direction,  are  those  of  a 
body  projected  vertically  upward  with  an  initial  velocity, 
V  sina,  and  then  continually  acted  on  by  the  force  of  gravity. 
Hence,  equation  (58)  is  applicable.  Making,  in  that  equa- 
tion, h  =  y,  and  v'  =  y  sina,  we  have, 

y  =  V  sina  i  ^  ^gf (87) 

Denote  the  space  described  in  the  direction  of  the  axis, 
A  By  in  the  time  /,  by  x.  The  only  force  acting  in  this 
direction  is  the  component  of  the  initial  impulse.  Hence, 
the  motion  will  be  uniform,  and  the  first  equation  of  Art. 
102,  is  applicable.    Making  s  =  x,  and  v  =  v  cosa,  we  have, 

X  =  V  COSa  t (88) 

If  we  suppose  t  to  be  the  same  in  equations  (87)  and  (88), 
they  will  be  simultaneous,  and,  taken  together,  will  make 
known  the  position  of  the  projectile  at  any  instant. 

From  (88),  we  have. 


t  = 


which,  substituted  in  (87),  gives, 

y==^a:--^— (89) 

an  equation  which  is  independent  of  /.  It,  therefore, 
expresses  the  relation  between  x  and  y  for  any  value  of  /, 
and  is,  consequently,  the  equation  of  the  trajectory.  But, 
equation  (89)  is  the  equation  of  a  parabola  whose  axis  is 
vertical.     Hence,  the  tnijectory  is  a  parabola. 

To  find  the  range,  make  y  -  0,  in  (89),  and  deduce  the 


CURVILINEAR   ANJ)    ROTARY    MOTION.  15*J 

corresponding  value  of  x.  Pliicing  the  value  of  y  equal  to  0, 
we  have, 

sina  (jx'      _ 

COSa  2v  cos  a 

,             2?;^sina  cosa 
.'.   a;  =  0,   and   x  ■= . 

The  first  value  of  x  corresponds  to  the  point  of  projec- 
tion, and  the  second  is  the  value  of  the  range,  AB, 
From  trigonometry,  we  have, 

2sina  cosa  =  sin2a. 

If  we  denote  the  height  due  to  the  initial  velocity,  by  /i, 
we  have, 

v"  =  Igli, 

Substituting  these  in  the  second  value  of  x,  and  denoting 
the  range  by  r,  we  have, 

r  =  2h  sin2a (90) 

The  greatest  value  of  r  will  correspond  to  a  =  45°,  in 
which  case,  2*  =  90°,  and  sin  2a  =  1.  Hence,  we  have, 
for  the  greatest  range, 

r  =  2h. 

That  is,  it  is  equal  to  iiuice  the  heigJU  due  to  the  initial 
velocity. 

If,  in  (90),  we  replace  a  by  90°  —  a,  we  have, 

r  =  2h  sin  (180°  —  2a)  =  2h  sin2a, 

the  same  value  as  before.  Hence,  there  are  two  angles  of 
prqiection,  complements  of  each  other,  that  give  the  same 
range.  The  trajectories  in  the  two  cases  are  not  the  same, 
as  may  be  shown  by  substituting  the  values  of  a,  and 
90°  —  a,  in  equation  (89).  Tlie  greater  angle  of  projection 
gives  a  higher  elevation,  and,  consequently,  the  projectile 


IGO 


MECHANICS. 


descends  more  vertically.  It  is  for  this  reason  that  tlie 
gunner  selects  the  greater  of  the  two,  when  he  desires  to 
crush  an  ohject,  and  the  less  when  lie  desires  to  batter,  or 
overturn  the  object.  If  a  =:  90°,  the  value  of  r  is  0. 
That  is,  if  a  body  be  projected  vertically  upward,  it  will 
return  to  the  point  of  projection. 

To  find  the  time   of  flight,  make   x  —  r,  in  (88),    and 
deduce  the  corresponding  value  of  t.     This  gives, 


t  = 


V  COSa 


(91) 


The  range  remaining  the  same,  the  time  of  flight  will  be 
greatest  when  a  is  greatest.  Equation  (88)  also  gives  the 
time  required  for  the  body  to  describe  any  distance  in  the 
direction  of  the  horizontal  line,  AB. 

In  equation  (91)  there  are  four  quantities,  t,  r,  v,  and  a, 
and  from  it,  if  any  three  are  given,  the  remaining  one  may 
be  determined. 

As  an  application  of  the  principles  just  deduced,  let  it 
be  required  to  determine  the  angle  of  projection,  that  the 
projectile  may  strike  a  point,  //, 
at  a  horizontal  distance,  AG  —  x' 
from  the  point  of  projection,  and 
at  a  height,  GH  =  y',  above  it. 

Since  H  lies  on  the  trajectory, 
its  co-ordinates  must  satisfy  the 
equation  of  the  curve,  giving, 


Fig.  102 


y'  =  x'  tana  - 

From  trigonometry,  we  have, 

1 
cos  a  =  5-  = 


gx 


Wgo&^cl 


1 


1  +  tanV 


CURVILINEAR   AND    ROTARY   3I0TI0N.  161 

Substituting  this  in  the  preceding  equation,  we  have, 
after  clearing  of  fractions, 

'Zv^y'  =  2i;Vtana  —  gx"^{l  +  tanV) ; 
or,  transposing  and  reducing. 


tan  a J  tana  = — ^ — 

gx  gx 


Hence, 


tana  =  — ;  =fc  y  — -^ j^ ; 

gx'  g^x'^  gx"" 


or,  making  v^  =  '2gh, 

tana  =  — ;-  db  y  -— '-^ = -, ^ . 

XX  X  X 

This  shoAvs  that  there  are  two  angles  of  projection,  under 
either  of  wliich,  the  point  may  be  struck. 
If  we  suppose 

x"  =  W  -  Ug' (92) 

the  quantity  under  the  radical  sign  will  be  0,  and  the  two 
angles  of  projection  will  become  one. 

If  x'  and  y'  be  regarded  as  variables,  equation  (92)  rep- 
resents a  parabola  whose  axis  is  a  vertical,  through  the 
point  of  projection.  Its  vertex  is  at  a  distance,  /z,  above 
the  point,  A,  its  focus  is  at  A,  and  its  parameter  is  4A,  or 
twice  the  range. 

If  we  suppose 

x"  <  4/i'  -  4:hy', 

the  point  (x,  ?/'),  will  lie  within  the  parabola  just  described, 
the  quantity  under  the  radical  sign  will  be  positive,  and 
there  will  be  two  real  values  of  tana,  and,  consequently, 
two  angles  of  projection,  under  either  of  which  the  point 
may  be  struck. 


162  MECHANICS. 

If  we  suppose 

x"  >  4^»  -  AJiy\ 

the  point  {x\  y'),  will  be  without  this  parabola,  the  values 
of  tana  will  both  be  imaginary,  and  there  will  be  no  angle 
under  which  the  point  can  be  struck. 


Let  the  parabola  B'LB  represent  the  curve  whose  equa- 
tion is 

x""  =  W  -  Ahy'. 

Conceive  it  to  be  revolved  about  AL^  as  an  axis,  gener- 
ating a  paraboloid  of  revolution.  Then,  from  what  pre- 
cedes, we  conclude,^r5^,  that  every  point  within  the  surface 
may  be  reached  from  A,  under  two  different  angles  of  pro- 
jection ;  secondly,  that  every  point  on  the  surface  can  be 
reached,  but  only  by  a  single  angle  of  projection ;  third- 
ly, that  no  point  without  the  surface  can  be  reached 
at  all. 

If  a  body  be  projected  horizontally  from  an  elevated 
point.  A,  its  trajectory  will  be  made  known  by  equation 
(89),  simply  making  a  =  0  ;  whence, 
sina  =  0,  and  cosa  =  1.  Substituting  and 
reducing,  we  have, 


(93) 


For   every   value   of  x,  y  is   negative, 


Fig.  104. 


which  shows  that  the  trajectory  lies  below  the  horizontal 


CUKVILIKEAR   AND    ROTARY   MOTION.  163 

through  the  point  of  projection.     If  we  suppose  ordinates 
to  be  positive  downward,  we  have, 

y=i (94) 

To  find  the  point  at  which  the  trajectory  will  reach  any 
horizontal  plane,  BC\  whose  distance  below  A  is  h',  make 
y  —  h',  in  (94),  whence, 

x  =  BC  =  vy— (95) 

On  account  of  the  resistance  of  the  air,  the  results  of 
the  preceding  discussion  must  be  greatly  modified.  They 
approach  more  nearly  to  the  observed  phenomena,  as  the 
velocity  is  diminished  and  the  density  of  the  projectile 
increased.  The  atmospheric  resistance  increases  as  the 
square  of  the  velocity,  and  as  the  cross  section  of  the  pro- 
jectile exposed  to  the  action  of  the  resistance.  In  the  air, 
it  is  found,  under  ordinary  circumstar.ces,  that  the  maxi- 
mum range  is  obtained  by  an  angle  of  projection,  not  far 
from  34°. 

Examples. 

1.  What  is  the  time  of  flight  of  a  projectile  in  vacuum,  wheu  the 
angle  of  projection  is  45°,  and  the  range  COOO  feet? 

SOLUTION. 

When  the  angle  of  projection  is  45°,  the  range  is  equal  to  twice  the 
height  due  to  the  velocity  of  projection.  Denoting  this  velocity  by 
V,  we  have, 

v''  =  2gh  =  2  X  32^  X  3000  =  193000. 
Whence, 

V  =  439.3  ft. 
From  equation  (91),  we  have, 

t  = =  -r— — —-^  =  19.3  sec.  Ans. 

vcosa      439.3  cos45 


164  MECHANICS. 

2.  What  is  the  range  of  a  projectile,  when  the  angle  of  projection 
is  30°,  and  the  initial  velocity  200  feet?  Ans.  1076.9  ft. 

3.  The  angle  of  projection  under  which  a  shell  is  thrown  is  32°, 
and  the  range  3250  feet.     What  is  the  time  of  flight? 

Ans.  11.25  sec,  nearly. 

Centripetal  and  Centrifugal  Forces. 

126.  Curvilinear  motion  can  only  result  from  the  action 
of  an  incessant  force,  whose  direction  differs  from  that  of 
the  original  impulse.  This  force  may  arise  from  one  or 
more  active  forces,  or  it  may  result  from  the  resistance 
offered  by  a  rigid  body,  as  when  a  ball  is  compelled  to  run 
in  a  groove.  Whatever  may  be  the  nature  of  the  forces, 
we  can  always  conceive  them  to  be  replaced  by  a  single 
incessant  force  acting  transversely  to  the  path  of  the  body. 
Let  this  force  be  resolved  into  two  components,  one  normal 
to  the  path  of  the  body,  and  the  other  tangential  to  it.  The 
latter  force  may  act  to  accelerate,  or  to  retard  the  motion 
of  the  body,  according  to  the  direction  of  the  resultant 
force ;  the  former  alone  is  effective  in  changing  the  direc- 
tion of  motion.  The  normal  component  is  always  directed 
toward  the  concave  side  of  the  curve,  and  is  called  the 
centripetal  force.  The  body  resists  this  force,  by  virtue  of 
its  inertia,  and,  from  the  law  of  inertia,  this  resistance 
must  be  equal  and  directly  opposed  to  the  centripetal  force. 
This  resistance  is  called  the  centrifugal  force.  Hence,  we 
may  define  centrifugal  force  to  be  the  resistance  a  body 
offers  to  a  force  that  tends  to  deflect  it  from  a  rectilineal 
path.  The  centripetal  and  centrifugal  forces  together,  are 
called  central  forces. 

Measiire  of  the  Centrifugal  Force. 

127.  To  deduce  an  expression  for  the  measure  of  the 
centrifugal  force,  let  us  first  consider  the  case  of  a  material 


CURVILINEAR   AND    ROTARY    MOTION. 


165 


point,  constrained  to  move  in  a  circular  path,  by  a  force 
constantly  directed  toward  the  centre,  as  when  a  body  is 
confined  by  a  string  and  whirled  around  a  fixed  point.  In 
this  case,  the  tangential  component  of  the  deflecting  force 
is  0 ;  there  is  no  loss  of  velocity  in  consequence  of  a 
change  of  direction  in  the  motion,  (Art.  109) ;  hence,  the 
motion  of  the  point  is  uniform. 

Let  ABD  be  the  path  of  the  body,  and  V  its  centre. 
Suppose  the  circumference  of  the  circle  to  be  a  regular 
polygon,  having  an  infinite  number  of 
sides,  of  which  AB  is  one;  and  denote 
each  side  by  s.  When  the  body  reaches 
A,  it  tends,  by  virtue  of  its  inertia,  to 
move  in  the  direction  of  the  tangent, 
A  T;  but,  in  consequence  of  the  action 
of  the  centripetal  force  directed  to- 
ward V,  it  is  constrained  to  describe 
the  side  s  in  the  time  t.  If  we  draw 
BC  parallel  to  A  T,  it  will  be  perpendicular  to  the  diameter 
AD,  and  ^  C  will  represent  the  space  through  w^hich  the 
body  has  been  drawn  from  the  tangent,  in  the  time  t  If 
we  denote  the  acceleration  due  to  the  centripetal  force  by 
/,  and  suppose  it  to  be  constant  during  the  time  t,  we 
have,  from  Art.  103, 


Ac=^fe 


(96) 


From  the  right-angled  triangle,  ABD,  we  have,  since 
AB  =  s, 

s'  =  ACx  AD;  or,  s'  =  AC  X  2r. 

Whence, 

2r 


166  MECHANICS. 

Substituting  this  value  of  AC,  in  (96),  and  solving  with 
respect  to/, 

•^        t        r 
But  TT  =  v"  (Art.  102),  in  which  v  is  the  velocity  of  the 

moving  point.      Substituting  in  the  preceding  equation, 
we  have, 

/=! (97) 

Here/  is  the  acceleration  due  to  the  centripetal  force, 
but  this  is  equal  to  the  centrifugal  force,  hence,  the  accel- 
eration due  to  the  centrifugal  force,  is  equal  to  the  square 
of  the  velocity,  divided  by  the  radius  of  the  circle. 

If  the  mass  of  the  body  be  denoted  by  M,  and  the  entire 
centrifugal  force  by  F,  we  have,  (Art.  18), 

F=^ (98) 

If  we  suppose  the  body  moving  on  any  curve,  we  may, 
whilst  it  is  passing  over  any  two  consecutive  elements,  re- 
gard it  as  moving  on  the  arc  of  the  osculatory  circle  to  the 
curve;  and,  further,  we  may  regard  the  velocity  as  uniform 
during  the  infinitely  small  time  required  to  describe  these 
elements.  The  direction  of  the  centrifugal  force  being 
normal  to  the  curve,  must  pass  through  the  centre  of  the 
osculatory  circle.  Hence,  all  the  circumstances  of  motion 
are  the  same  as  before,  and  equations  (97)  and  (98)  will  be 
applicable,  provided  r  be  taken  as  the  radius  of  the  cun^a- 
ture.  Hence,  Ave  may  enunciate  the  law  of  centrifugal 
force  as  follows : 

The  acceleration  due  to  the  centrifugal  force  is  equal  to 
the  square  of  the  velocity  of  the  body  divided  hy  the  radius 
of  curvature. 


CURVILINEAR   AND    ROTARY    MOTION. 


167 


The  entire  centrifugal  force  is  equal  to  the  acceleration, 
multiplied  hy  the  7nass  of  the  body. 

Ill  the  case  of  a  body  whirled  around  a  centre,  and  re- 
strained by  a  string,  the  tension  of  the  string  is  measured 
by  the  centrifugal  force.  The  radius  remaining  constant, 
the  tension  increases  as  the  square  of  the  velocity. 


Centrifugal  Force  at  points  of  the  Earth's  Surface. 

128.  Let  it  be  required  to  determine  the  centrifugal 
force  at  different  points  of  the  earth's  surface,  due  to  rota- 
tion on  its  axis. 

Suppose  the  earth  spherical.  Let  ^  be  a  point  on  the 
surface,  PQP'  a  meridian  section  through  A,  PP'  the  axis, 
RQ  the  equator,  and  J^,  per- 
pendicular to  PP',  the  radius 
of  the  parallel  of  latitude 
through  A.  Denote  the  radius 
of  the  earth  by  r,  the  radius 
of  the  parallel  through  A  by 
r',  and  the  latitude  of  ^,  or 
the  angle  ACQ,  by  I.  The 
time  of  revolution  being  the 
same  for  every  point  on  the 
earth's  surface,  the  velocities  of  Q  and  A  will  be  to  each 
other  as  their  distances  from  the  axis.  Denoting  these  velo- 
cities by  V  and  v',  we  have, 


whence, 


V  :  V 


V    = 


vr 


But  from   the   right-angled   triangle,  CAB,   since  the 
angle  at  A  is  equal  to  /,  we  have, 

r'  =  r  cosZ. 


168  MECHANICS. 

Substituting  this  value  of  r'  in  the  value  of  v',  and  re- 
ducing, we  have, 

v'  —  V  cos/. 

If  we  denote  the  centrifugal  force  at  the  equator  by/, 
we  have, 

f=j (99) 

In  like  manner,  if  we  denote  tlie  centrifugal  force  at  A^ 
by/',  we  have, 

r 

Substituting  for  v'  and  r'  their  values,  previously  de- 
duced, we  get, 

/'  =  ^' (100) 

Combining  equations  (99)  and  (100),  we  find, 
/  :  /'  :  :  1  :  cos/,         .'.    /'  =/cos/ (101) 

That  is,  the  centrifugal  force  at  any  point  on  the  earth's 
surface,  is  equal  to  the  centrifugal  force  at  the  equator, 
multiplied  hy  the  cosine  of  tlie  latitude. 

Let  AE,  perpendicular  to  PP',  represent/',  and  resolve 
it  into  two  components,  one  tangential,  and  the  other 
normal  to  the  meridian  section.  Prolong  CA,  and  draw 
AD  perpendicular  to  it  at  ^.  Complete  the  rectangle,  FD 
on  AB,  a.s  a  diagonal.  Then  will  AD  be  the  tangential, 
and  AF  the  normal  component.  In  the  right-angled 
triangle,  AFF,  the  angle  at  A  is  equal  to  L     Hence, 

FF=AD^f'sml  =  fco8lsml  =  ^^^— (102) 

AF  =  f  cost  =fcos'l (103) 

From  (102),  we  see  that  the  tangential  component  is  0 
at  the  equator,  goes  on  increasing  till  /  =  45°,  where  it  is  a 


CURVILINEAR   AND    ROTARY    MOTION.  169 

maximum,  and  then  goes  on  decreasing  till  the  latitude  is 
90°,  when  it  again  becomes  0. 

The  effect  of  the  tangential  component  is  to  heap  up  the 
particles  of  the  earth  about  the  equator,  and,  were  the 
earth  in  a  fluid  state,  this  process  would  go  on  till  the  effect 
of  the  tangential  component  was  counterbalanced  by  the 
component  of  gravity  acthig  down  the  inclined  plane 
thusformed,  when  the  particles  would  be  in  equilibrium. 

The  higher  analysis  shows  that  the  form  of  equilibrium 
is  that  of  an  oblate  spheroid,  differing  but  slightly  from 
that  which  our  globe  is  found  to  possess  by  actual  measure- 
ment. 

From  equation  (103),  we  see  that  the  normal  component 
of  the  centrifugal  force  varies  as  the  square  of  the  cosine 
of  the  latitude. 

This  component  is  directly  opposed  to  gravity,  and,  con- 
sequently, tends  to  diminish  the  apparent  weight  of  all 
bodies  on  the  surface  of  the  earth.  The  value  of  this  com- 
ponent is  greatest  at  the  equator,  and  diminishes  toward 
the  poles,  where  it  is  0.  From  the  action  of  the  normal 
component  of  the  centrifugal  force,  and  because  the  flat- 
tened form  of  the  earth  due  to  the  tangential  component 
brings  the  polar  regions  nearer  the  centre  of  the  earth,  the 
measured  force  of  gravity  ought  to  increase  in  passing  from 
the  equator  toward  the  poles.  This  is  found  to  be  the 
case. 

The  radius  of  the  earth  at  the  equator  is  about  3962.8 
miles,  which,  multiplied  by  S-tt,  will  give  the  entire  circum- 
ference of  the  equator.  If  this  be  divided  by  the  number 
of  seconds  in  a  day,  86400,  we  find  the  value  of  v.  Substi- 
tuting this  value  of  v  and  that  of  r  just  given,  in  equation 
(99),  we  find, 

f=  0.1112  ft., 


170  MECHANICS. 

for  the  centrifugal  force  at  the  equator.  If  this  be  multi- 
plied by  the  square  of  the  cosine  of  the  latitude  of  any 
place,  we  have  the  value  of  the  normal  component  of  the 
centrifugal  force  at  that  place. 

Centrifugal  Force  of  Extended  Masses. 

129.  We  have  supposed,  in  what  precedes,  the  dimen- 
sions of  the  body  under  consideration  to  be  extremely  small ; 
let  us  next  examine  the  case  of  a  body,  of  any  dimensions 
whatever,  constrained  to  revolve  about  a  fixed  axis.  If  the 
body  be  divided  into  infinitely  small  elements,  whose  di- 
rections are  parallel  to  the  axis,  the  centrifugal  force  of 
each  element  Avill  be  equal  to  the  mass  of  the  element  into 
the  square  of  its  velocity,  divided  by  its  distance  from  the 
axis.  If  a  plane  be  passed  through  the  centre  of  gravity 
of  the  body,  perpendicular  to  the  axis,  we  may,  without 
impairing  the  generality  of  tlie  result,  suppose  the  mass  of 
each  element  concentrated  at  the  point  in  which  this  plane 
cuts  the  line  of  direction  of  the  element. 

Let  XCY  be  the  plane  through  the  centre  of  gravity 
perpendicular  to  the  axis  of  revolution,  AB  the  projection 
of  the  body  on  the   plane,  and  C  the  — -^^ 

point  in  which  it  cuts  the  axis.  Take 
C  as  the  origin  of  a  system  of  rectan- 
gular co-ordinates ;  let  OJC  be  the  axis 
of  X,  CY  the  axis  of  Y,  and  m  be  the 
point  at  which  the  mass  of  one  filament     ^  ^ 

is  concentrated,  and  denote  that  mass  ^^^'  ^^^' 

by  m.  Denote  the  co-ordinates  of  w  hjx  and  y,  its  dis- 
tance from  C  by  r,  and  its  velocity  by  v.  The  centrifugal 
force  of  the  mass,  m,  is  equal  to 

r 


CURVILINEAR    AND    ROTARY    MOTION.  171 

If  we  denote  the  angular  velocity  of  the  body  by  V,  the 
velocity  of  m  will  be  r  V,  which,  in  the  expression  just 
deduced,  gives, 

mr  V'\ 

Let  this  force  be  resolved  into  components  parallel  to 
GX  and  CY.     We  have,  for  these  components, 

mr  V  "cosm  OX,    and,    7nr  V  '^simn  CX, 

But,  from  the  figure, 

G0S7nCX  — —,   and,   smmCX=K 
r  r 

Substituting  these  in  the  preceding  expressions,  and 
reducing,  we  have,  for  the  components, 

mxV''\     and,    myV"^. 

Similar  expressions  may  be  deduced  for  each  of  the  other 
filaments.  If  we  denote  the  resultant  of  the  components 
parallel  to  CX  by  X,  and  of  those  parallel  to  CY  by  Y, 
we  have, 

X=i:{mx)V'\    and,    Y:=^I(my)V'\ 

If  we  denote  the  mass  of  the  body  by  if,  and  suppose  it 
concentrated  at  its  centre  of  gravity,  0,  whose  co-ordi- 
nates are  x^,  and  y^,  and  whose  distance  from  C  is  7\,  we 
shall  have,  from  the  principle  of  the  centre  of  gravity, 
(Art.  55), 

I{mx)  =  Mx^,  and  ^{my)  =  My^. 

Substituting  above,  we  have, 

X^MV^'x^,    and,     F^rJfF'y. 

If  we  denote  the  resultant  centrifugal  force  by  R,  we 
have. 


R  =  VX'  4-  Y'  =  MV'WK  +  y"  =  ^V'\^. 


172  MECHANICS. 

But  if  the  vel()(;ity  of  the  centre  of  gravity  be  denoted 
by  F,  we  have, 

r=  FV.;    or,    F"  =  ^  ; 

'  1 

which,  in  the  preceding  result,  gives, 

R  =  ^ (104) 

The  direction  of  R  is  given  by  the  equations, 

cos  a  =  -77  =  -,     and    cos  £>  = -77- =  — ; 

hence,  it  passes  through  the  centre  of  gravity,  0 ;  that  is, 
the  centrifugal  force  of  an  extended  mass,  constramed  to 
revolve  about  a  fixed  axis,  is  the  same  as  though  the  mass 
luere  concentrated  at  its  centre  of  gravity. 


Principal  Axes. 

130.  Suppose  the  axis  about  which  a  body  is  revolving 
to  be  free,  so  that  the  body  can  move  in  any  manner.  If 
the  body  is  homogeneous  and  the  axis  not  one  of  symme- 
try, the  centrifugal  forces  of  the  elements  of  the  body  will 
not  balance  each  other,  and  unequal  pressures  will  be 
exerted  on  different  parts  of  the  axis.  This  inequality  of 
pressure  will  change  the  position  of  the  axis  of  revolution 
at  each  instant,  and  the  change  will  go  on,  till  an  axis  is 
reached,  that  is  pressed  equally  in  all  directions  by  the  cen- 
trifugal forces  of  the  elements.  Such  an  axis  is  called  a 
principal  axis.  It  may  be  shown,  by  the  higher  analysis, 
that  a  body  has  at  least  tliree  principal  axes,  which  pass 
through  its  centre  of  gravity,  and  are  at  right  angles  to 
each  other.  It  may  also  be  shown  that  the  moment  of 
inertia  with  respect  to  one  of  these  axes  is  greater,  and 


CUKVILINEAR   AKD    ROTARY    MOTIOI^.  173 

with  respect  to  another  less,  than  with  reference  to  any 
other  line  through  the  centre  of  gravity.  When  the  body 
is  revoh-iftg  about  the  former,  its  rotation  is  stable;  when 
about  the  latter,  it  is  unstable.  The  former  may  be  called 
an  axis  of  stability,  and  the  latter  an  axis  of  instability. 
In  tlie  case  of  certain  regular  bodies,  there  may  be  an  infi- 
nite number  of  either  kind.  Thus,  in  an  oblate  spheroid, 
the  polar  axis  is  an  axis  of  stability,  and  the  only  one, 
whilst  any  diameter  of  an  equatorial  section  is  an  axis  of 
instability.  In  a  prolate  spheroid,  the  polar  axis  is  an  axis 
of  instability,  and  the  only  one,  whilst  any  diameter  of  the 
equatorial  section  is  an  axis  of  stability.  In  a  right  cone 
with  a  circular  base,  the  axis  of  the  cone  is  an  axis  of 
instability;  but  any  line  through  the  centre  of  gravity,  and 
perpendicular  to  the  axis,  is  an  axis  of  stability. 

Experimental  Illustrations. 

131.  The  principles  relating  to  centrifugal  force  admit 
of  experimental  illustration.  The  instrument  represented 
in  the  figure  may  be  employed  to  _ 

show  the  value  of  the  centrifugal    ^  u56^S^ 
force.     A   is   a   vertical   axle,    on 


E  D 

C 


which  is  mounted  a  wheel,  F.  com-  a 


lA 


municating  with  a  train  of  wheel-  |j|| 

work,  by  means  of  which  the  axle  PJg-  los. 

may  be  made  to  revolve  with  any  angular  velocity.  At 
the  upper  end  of  the  axle  is  a  forked  branch,  BC,  sustain- 
ing a  stretched  wire.  D  and  E  are  balls  pierced  by  the 
wire,  and  free  to  move  along  it.  Between  B  and  ^  is  a 
spiral  spring,  whose  axis  coincides  with  the  wire. 

Immediately  below  the  spring,  on  the  horizontal  part  of 
the  fork,  is  a  scale  for  determining  the  distance  of  the  ball, 
E,  from  the  axis,  and  for  measuring  the  degree  of  compres- 


174  ^         MECHANICS. 

sion  of  the  spring.  Before  using  the  instrument,  the  force 
required  to  produce  any  degree  of  compression  of  the 
spring  is  determined  experimentally,  and  marked  on  tlie 
scale. 

If  a  motion  of  rotation  be  communicated  to  the  axis, 
the  ball  D  will  at  once  recede  to  C,  but  the  ball  E  will  be 
restrained  by  the  spring.  As  the  velocity  of  rotation 
increases,  the  spring  is  compressed  more  and  more,  and  the 
ball  E  approaches  B.  By  a  suitable  ari-angement  of  wheel- 
work,  the  angular  velocity  of  the  axis  corresponding  to 
any  compression  may  be  ascertained.  We  have,  therefore, 
all  the  data  necessary  to  verify  the  law  of  centrifugal 
force. 

If  a  vessel  of  water  be  made  to  revolve  about  a  vertical 
axis,  the  inner  particles  recede  from  the  axis  on  account 
of  the  centrifugal  force,  and  are  heaped  up  about  the  sides 
of  the  vessel,  imparting  a  concave  form  to  the  upper  sur- 
face. The  concavity  becomes  greater  as  the  angular  velo- 
city is  increased. 

If  a  circular  hoop  of  flexible  material  be  mounted  on 
one  of  its  diameters,  its  lower  point  being  fastened  to  the 
horizontal  beam,  and  a  motion  of  rotation  imparted,  the 
portions  of  the  hoop  farthest  from  the  axis  will  be  most 
aff*ected  by  centrifugal  force,  and  the  hoop  will  assume  an 
elliptical  form. 

If  a  sponge,  filled  with  water,  be  attached  to  one  of  the 
arms  of  a  whirling  machine,  and  motion  of  rotation  im- 
parted, the  water  will  be  thrown  from  the  sponge.  This 
principle  has  been  used  for  drying  clothes.  An  annular 
trough  of  copper  is  mounted  on  an  axis  by  radial  arms, 
and  the  axis  connected  with  a  train  of  wheelwork,  by 
means  of  which  it  may  be  put  in  motion.  The  outer  wall 
of  the  trough  is  pierced  with  holes  for  the  escape  of  water, 


OUJtVILINEAR   AND   ROTARY   MOTION^.  175 

and  a  lid  confines  tlie  articles  to  be  dried.  To  use  this 
instrument,  the  linen,  after  being  washed,  is  placed  in 
the  annular  space,  and  a  rapid  rotation  imparted  to  the 
machine.  The  linen  is  thrown  against  theouter  wall  of 
the  instrument,  and  the  water,  urged  by  the  centrifugal 
force,  escapes  through  the  holes.  Sometimes  as  many  as 
1,500  revolutions  per  minute  are  given  to  the  drying  ma- 
chine, in  which  case,  the  drying  process  is  very  rapid  and 
very  perfect. 

If  a  body  revolve  with  sufficient  velocity,  it  may  happen 
that  the  centrifugal  force  generated  will  be  greater  than 
the  force  of  cohesion  that  binds  the  particles  together,  and 
the  body  be  torn  asunder.  It  is  a  common  occurrence  for 
large  grindstones,  when  put  into  rapid  rotation,  to  burst, 
the  fragments  being  thrown  away  from  the  axis,  and  often 
producing  much  destruction. 

When  a  wagon,  or  carriage,  is  driven  round  a  corner,  or 
is  forced  to  run  on  a  circular  track,  the  centrifugal  force  is 
often  sufficient  to  throw  loose  articles  from  the  vehicle,  and 
even  to  overthrow  the  vehicle  itself  When  a  car  on  a  rail- 
road track  is  forced  to  turn  a  sharp  curve,  the  centrifugal 
force  throws  the  cars  against  the  rail,  producing  a  great 
amount  of  friction.  To  obviate  this  difficulty,  it  is  cus- 
tomary to  raise  the  outer  rail,  so  that  the  resultant  of  the 
centrifugal  force,  and  the  force  of  gravity,  shall  be  perpen- 
dicular to  the  plane  of  the  rails. 

Elevation  of  the  Outer  Rail  of  a  Curved  Track. 

132.  To  find  the  elevation  of  the  outer  rail,  so  that 
the  resultant  of  the  weight  and  centrifugal  force  shall 
be  perpendicular  to  the  line  joining  the  rails,  assume 
a  cross  section  through  the  centre  of  gravity,  O.     Take 


176 


MECHANICS. 


the  horizontal,  GA,  to  represent 
the  centrifugal  force,  and  GB  to 
represent  gravity.  Construct 
their  resultant,  G  C.  Then  must 
DEhe  perpendicular  to  GC 

Denote  the  velocity  of  the  car 
by  V,  the  radius  of  the  curved 
track  by  r,  the  force  of  gravity  Fig.  109. 

by  g,  and  the  angle,  DEF,  or  its  equal,  BGC,hj  a.     From 
the  right-angled  triangle,  GBO,  we  have, 


tana  = 


BO 
GB' 


But,  BC,  or  its  equivalent,  GA,  is  equal  to  — ,  and  GB  is 


equal  to  g  ;  hence. 


tana  = 


gr 


Denoting  the  distance  between  the  rails,  by  d,  and  the 
elevation  of  the  outer  rail  above  the  inner  one,  by  h,  we 
have, 

tana  =  — ,  very  nearly. 
Equating  the  two  values  of  tana,  we  have, 

d~  gr'    '  '     ^  ~  gr' 

Hence,  the  elevation  of  the  outer  rail  varies  as  the  square 
of  the  velocity  directly,  and  as  the  radius  of  the  curve 
inversely. 

It  is  obvious  tliat  the  elevation  ought  to  be  different  for 
different  velocities,  which,  from  the  nature  of  the  case,  is 
impossible.  The  correction  is,  therefore,  made  for  some 
assumed  velocity,  and  then  such  a  form  is  given  to  the 


CURVILli^^EAR    AND    ROTAIIY    .MOTIOX.  177 

tire  of  the  wheels  as  will  complete  the  correction  for  other 
velocities. 

The  Conical  Pendulum. 

133.  The  conical  pendulum  consists  of  a  ball  attached 
to  one  end  of  a  rod,  the  other  end  of  which  is  connected, 
by  a  hinge-joint,  with  a  vertical  axle.  When  the  axle  is 
put  in  motion,  the  centrifugal  force  causes  the  ball  to 
recede  from  the  axis,  until  an  equilibrium  is  established 
between  the  weight  of  the  ball,  the  centrifugal  force,  and 
the  resistance  of  the  connecting  rod.  When  the  velocity  is 
constant,  the  centrifugal  force  is  constant,  and  the  centre  of 
the  ball  describes  a  horizontal  circle,  whose  radius  depends 
on  the  velocity.     To  determine  the  time  of  revolution  : 

Let  BD  be  the  axis,  A  the  ball,  B  the  hinge-joint,  and 
AB  the  connecting  rod,  whose  mass  is  so  small,  that  it 
may  be  neglected,  in  comparison  with  that  -g 
of  the  ball. 

Denote  the  time  of  revolution,  by  t,  the 
length  of  the  arm,  by  I,  the  centrifugal 
force,  by  /,  and  the  angle,  ABC,  by  9. 
Draw  A  C  perpendicular  to  BD,  and  denote 
^(7,  by  r,  and^(7,  by7^.  <''' 

From    the     triangle,    ABC,    we    have,  '"' 

r  =  h  tancp;  and  since  r  is  the  radius  of  the  circle  described 
by  A,  the  distance  passed  over  by  A,  in  the  time  t,  is  equal 
to  'Z'Ttr,  or,  2'7r/itan9.  Denoting  the  velocity  of  A,  by  v,  we 
have,  from  article  (102), 

S^r/^tana) 

But  the  centrifugal  force  is  equal  to  the  square  of  the 
velocity,  divided  by  the  radius ;  hence, 

/=^^ (105) 


178 


MECHANICS. 


The  /orces  that  act  on  A,  are  the  centrifugal  force,  in 
the  dir<;ction  AF,  the  force  of  gravity,  in  the  direction  AG, 
and  the  resistance  of  the  connecting  rod,  in  the  direction 
AB.  In  order  that  the  ball  may  remain  at  an  invariable 
distance  from  the  axis,  these  must  be  in  equilibrium. 
Hence,  (Art.  33), 

g  :  f  ::  sinBAF  ::    s'mBAG; 
but,  m\BAF=  sin(90°  +  cp)  =  cos(p; 

and,  sin^^6^  =  sin(180°  —  qp)  —  s'mcp. 

We  have,  therefore, 

g  •  /  •  •  cos(p  :  sin(p,     .*.  f  =  g  tan^. 
Equating  these  values  of  f,  we  have, 
4'7rV^  tanqj 


=  ^tan(p. 


Solving  with  respect  to  /, 

/  =  2^|/^ (106) 

That  is,  the  time  of  revolution,  is  equal  to  the  time  of 
a  double  vibration  of  a  pendulum  whose  length  is  h. 

The  Governor. 

134.  The  principle  of  the  conical  pendulum  is  employed 
in  the  governor,  a  machine  attached  to  engines,  to  regulate 
the  supply  of  motive  force. 

^^  is  a  vertical  axis  connected 
with  the  machine  near  its  Avorking- 
point,  and  revolving  witli  a  velocity 
proportional  to  tliat  of  the  working- 
point;  i^^^and  (W  are  arms  turning 
about  AB,  and  bearing  heavy  balls, 
D  and  E,  at  their  extremities ;  these 
bars  are  united  by  hinge-joints  with 


CURVILINEAR   AND    ROTARY    MOTION.  179 

two  other  bars  at  G  and  F,  and  also  to  a  ring  at  H,  that 
is  free  to  slide  up  and  down  the  shaft. 

The  ring,  H,  is  connected  with  a  lever,  HK,  that  acts 
on  the  valve  in  the  pipe  that  admits  steam  to  the  cylinder. 

When  the  shaft  revolves,  the  centrifugal  force  causes 
the  balls  to  recede  from  the  axis,  and  the  ring,  H,  is 
depressed ;  and  when  the  velocity  has  become  sufficiently 
great,  the  lever  closes  the  valve.  If  the  velocity  slackens, 
the  balls  approach  the  axis,  and  the  ring,  H,  ascends,  open- 
ing the  valve.  In  any  given  case,  if  Ave  know  the  velocity 
required  at  the  working-point,  we  can  compute  the  required 
angular  velocity  of  the  shaft,  and,  consequently,  the  value 
of  t.  This  value  of  /,  substituted  in  equation  (106),  gives 
the  value  of  h.  We  may,  therefore,  proDerly  adjust  the 
ring,  and  the  lever,  HK. 

Examples. 

1.  A  ball  weighing  10  lbs.  is  whirled  round  in  a  circle  whose  radius 
is  10  feet,  with  a  velocity  of  30  feet.  What  is  the  acceleration  due 
to  centrifugal  force  ?  Ans.  90  ft. 

2.  In  the  preceding  example,  what  is  the  tension  on  the  cord  that 
restrains  the  ball  ? 

SOLUTION. 

Denote  the  tension,  in  pounds,  by  t;  then,  since  the  pressures  pro- 
duced are  proportional  to  the  accelerations,  we  have, 

10  :  ^  :  :  5'  :  90,  .'.  ^  =  28  lbs.,  nearly.  Arm. 

3.  A  body  is  whirled  round  in  a  circular  path  whose  radius  is  5 
feet,  and  the  centrifugal  force  is  equal  to  the  weight  of  the  body. 
What  is  the  velocity  of  the  moving  body  ? 

SOLUTION. 

Denoting  the  velocity  by  «,  we  have  the  centrifugal  force  equal 
to  —  ;  but,  by  the  conditions  of  the  problem,  this  is  equal  to  gravity; 

hence,  —  =  32^ ;  or,  v  =12.7  ft.  Ans. 


180  MECHANICS. 

4.  lu  how  many  seconds  must  the  earth  revolve  that  tlie  centrif- 
ugal force  at  the  equator  may  counterbalance  the  force  of  gravity, 
the  radius  of  the  equator  being  3962.8  miles  V 

SOLUTION. 

Reducing  miles  to  feet,  and  denoting  the  required  velocity,  by  «, 
we  have. 


V  =  v/82^  X  20928584, 


20923584 

But  the  time  of  revolution  is  equal  to  the  circumference  of  the 
equator,  divided  by  the  velocity.     Denoting  the  time  by  t,  we  have, 
^  _  27r  X  20923584 

and,  substituting  for  «,  its  value,  taken  from  the  preceding  equation, 
we  have, 

,  _  2;r  \/20923584      27t  X  4574       ,^,., 

'  = 7= = ^-;^ =  5068  sees.  Ans. 

v/321  0.67 

5.  A  body  is  placed  on  a  horizontal  plane,  which  is  made  to  re- 
volve about  a  vertical  axis,  with  an  angular  velocity  of  2  feet.  How 
far  must  the  body  be  situated  from  the  axis  tliat  it  may  be  on  the 
point  of  sliding  outward,  the  coefficient  of  friction  between  the  body 
and  plane  being  equal  to  .6  ? 

SOLUTION. 

Denote  the  required  distance  by  r;  then  will  the  velocity  of  the 
body  be  2r,  and  the  centrifugal  force  4r.  But  the  acceleration  due 
to  the  force  of  friction  is  equal  to  0.6  X  g  =  19.3  ft.  From  the  con- 
ditions of  the  problem,  these  are  equal,  hence, 

4/'  =  19.3  ft.,  .-.  r  =  4.825  ft.  Ans. 

6.  What  must  be  the  elevation  of  the  outer  rail  of  a  track,  the 
radius  being  3960  ft.,  the  distance  between  the  rails  5  feet,  and  the 
velocity  of  the  car  30  miles  per  hour,  that  there  may  be  no  latera\ 
thrust?  Ans.  0.076  ft.,  or  0.9  in.,  nearly. 

7.  The  distance  between  the  rails  is  5  feet,  the  radius  of  the  curve 
600  feet,  and  the  height  of  the  centre  of  gravity  of  the  car  5  feet. 
What  velocity  must  the  car  have  that  it  may  be  on  the  point  of  being 
overturned  by  the  centrifugal  force,  the  rails  being  on  the  same  level? 

We  have. 


/5  X  32i  X  600      ,^^  -  ^^,  .  . 

V  =  \/ r L =  98  ft.,  or  66|  m.,  per  hour.  Ans. 

V  <  X  0 


CUKVlLlNEAIi   AND    ROTARY    MOTION.  181 


Definition  and  Measure  of  Work. 

135.  By  the  term  tvork,  in  mechanics,  is  meant  the  effect 
produced  by  a  force  in  overcoming  a  resistance.  It  implit'S 
that  a  force  is  exerted  through  a  certain  space  ;  thus,  a 
force  exerted  to  raise  a  weight  is  said  to  work,  and  the 
quantity  of  ivorh  performed  depends,  first  on  the  weight 
raised,  and  secondly  on  the  height  through  wliich  it  is 
raised.  Because  other  kinds  of  work  may  be  assimilated  to 
that  of  raising  a  weight,  it  is  customary  to  assume  tlie  work 
necessary  to  raise  a  given  weight,  to  a  given  height,  as  a 
standard  to  which  all  kinds  of  work  may  be  referred. 

In  this  country,  and  in  Great  Britain,  the  unit  generally 
adopted  is  the  work  required  to  raise  a  weight  of  onepoiind 
through  a  height  of  one  foot.  This  unit  is  called  ?i  foot- 
pound. In  France,  the  assumed  unit  is  the  work  required 
to  raise  a  kilogramme  through  a  metre ;  it  is  called  a  kilo- 
grammetre. 

If  we  denote  the  force  exerted  by  P,  the  space  through 
which  it  is  exerted  by  |?,  and  the  quantity  of  work  per- 
formed by  ft  we  shall  have, 

Q  =  Pv- 

If  the  force  is  variable,  we  may  conceive  the  path  divided 
into  equal  parts,  so  small  that,  for  each  part,  the  pressure 
may  be  regarded  as  constant.  If  we  denote  the  length  of 
one  of  these  parts  by  jo,  and  the  force  exerted  whilst  de- 
scribing it  by  jP,  we  shall  have,  for  the  corresponding  quan- 
tity of  work,  P;j,  and  for  the  entire  quantity  of  work, 
denoted  by  Q,  we  shall  have  the  sum  of  the  elementary 
quantities  of  work ;  or,  since  p  is  the  same  for  each. 

The  quotient  obtained  by  dividing  the  entire  quantity  of 


182  MECHANICS. 

work  by  the  entire  path,  is  called  the  mean  pressure,  or  the 
mean  resistance,  and  is  evidently  the  force  which,  acting 
uniformly  through  the  same  path,  would  accomplish  the 
same  work. 

In  estimating  work  performed  by  engines  and  other 
machines,  a  unit  is  adopted  that  involves  the  additional 
idea  of  time.  This  unit  is  called  a  horse  potver.  A  horse 
power  is  a  power  capable  of  raising  33,000  lbs.  through  a 
height  of  1  foot  in  1  minute.  When  we  say  that  a  machine 
is  one  of  10  horse  power,  we  mean  that  it  is  capable  of  per- 
forming 330,000  units  of  work  in  a  minute. 

Work,  when  the  Power   acts  Obliquely. 

136.  Let  PD  be  a  force,  and  AB  the  path  that  the  body 
D  is  constrained  to  follow.  Denote  the  angle  PDs  by  a, 
and  suppose  P  to  be  resolved  into  two 
components,  one  perpendicular,  and 
the  other  parallel  to  AB.  We  have,  a  ^  i>  B 
for  the  former,  Psina,  and,  for  the  lat-  ■^^^*  ^^^' 

ter,  Pcosft. 

The  former  can  produce  no  work,  since,  from  the  nature 
of  the  case,  the  point  cannot  move  in  the  direction  of  the 
normal ;  hence,  the  latter  is  the  only  component  that 
works.  Let  sD  be  the  space  through  which  the  body  is 
moved,  and  denote  the  quantity  of  work,  by  Q  ;  we  have, 

Q  =  Pcosa  X  sD. 

Let  fall  the  perpendicular  ss'  from  s,  on  the  direction  of 
the  force,  P.  From  the  right-angled  triangle,  Dss',  we  have, 

sD  X  cosa  =  s'D. 

Substituting  this  in  the  preceding  equation,  we  get, 

Q  =  PX  s'D, 


CURVILINEAR    AND    IIOTARY    MOTION.  183 

Tluit  is,  the  quantity  of  work  of  a,  force  acting  obliquely 
to  the  path  along  which  the  point  of  application  is  con- 
strained to  move,  is  equal  to  the  intensity  of  the  force  mul- 
tiplied by  the  projection  of  the  path  on  the  direction  of  the 
force. 

If  we  take  sD,  infinitely  small,  s'JJ  will  be  the  virtual 
velocity  of  D,  and  the  expression  for  the  quantity  of  work 
of  Pwill  be  its  virtual  moment,  (Art.  36).  Hence,  we  say 
that  the  elementary  quantity  of  work  of  a  force  is  equal  to 
its  virtual  moment,  and,  from  the  principle  of  virtual  mo- 
ments, we  conclude  that  the  algebraic  sum  of  the  element- 
ary quantities  of  work  of  any  number  of  forces  applied  at 
the  same  point,  is  equal  to  the  elementary  quantity  of  work 
of  their  resultant.  What  is  true  for  the  elementary  quan- 
tities of  work  at  any  instant,  must  be  equally  true  at  any 
other  instant.  Hence,  the  algebraic  sum  of  all  the  element- 
ary quantities  of  work  of  the  components  is  equal  to  the 
algebraic  sum  of  the  elementary  quantities  of  work  of 
their  resultant;  that  is,  the  work  of  the  components  is 
equal  to  the  work  of  their  resultant. 

This  principle  hardly  seems  to  require  demonstration, 
for,  from  the  definition  of  a  resultant,  it  Avould  seem  to  be 
true  of  necessity.  If  the  forces  be  in  equilibrium,  the 
entire  quantity  of  work  is  equal  to  0. 

This  principle  is  used  in  computing  the  quantity  of 
work  required  to  raise  material  for  a  wall  or  building ;  for 
raising  material  from  a  shaft ;  for  raising  water  from  one 
reservoir  to  another;  and  for  a  great  variety  of  similar 
operations.  In  this  connection,  the  principle  may  be  enun- 
ciated as  follows : 

The  algebraic  sum  of  the  quantities  of  work  required  to 
raise  the  parts  of  a  system  through  any  vertical  spaces,  is 
equal  to  the  quantity  of  work  required  to  move  the  whole 


184  MECHAKICS. 

system  through  the  height  described  hy  the  centre  of  gravity 
of  the  system. 

It  also  follows,  that,  if  atl  the  pieces  of  a  machine,  ivhich 
moves  without  friction,  he  in  equilihrinm  in  all  positions, 
the  centre  of  gravity  of  the  system  tvill  neither  ascend  nor 
€lescend  lohilst  the  machine  is  in  motion. 

Rotation. 

137.  When  a  body,  restrained  by  an  axis,  is  acted  on  by 
a  force  not  passing  through  the  axis,  it  takes  up  a  motion 
of  rotation.  In  this  variety  of  motion,  each  point  describes 
a  circle  whose  plane  is  perpendicular  to,  and  whose  centre 
is  in,  the  axis.  The  velocity  of  any  particle  is  equal  to  its 
distance  from  the  axis  multiplied  by  the  angular  velocity 
of  the  body.  The  time  of  revolution  of  all  the  particles 
being  the  same,  the  velocities  of  different  particles  are  pro- 
portional to  their  distances  from  the  axis. 

Quantity  of  Work  of  a  Force  producing  Rotatiom. 

138.  If  a  force  be  applied  obliquely  to  the  axis  of  rotation, 
we  may  conceive  it  to  be  resolved  into  two  components,  one 
parallel,  and  the  other  perpendicular  to  the  axis.  The 
effect  of  the  former  will  be  counteracted  by  the  resistance 
of  the  axis;  the  effect  of  the  latter  will  be  exactly  the 
same  as  that  of  the  applied  force.  We  need,  therefore, 
consider  those  components  only,  whose  directions  are  per- 
pendicular to  the  axis. 

Let  P  be  a  force  whose  direc-  t>  b 

tion  is  perpendicular  to  the  axis,       •. tv — y'^^'A. *^ 

but  does  not  intersect  it.     Let  0      \    // yf--^'^ 

be   the   point   in  which   a   plane  r^/{^'^' 
through  P,  perpendicular  to  the  ^     ^^^ 

axis,  intersects  it.     Let  A  and  C 


CURVILINEAR    AND    ROTARY    MOTION.  185 

be  any  two  points  on  the  direction  of  P.  Suppose  P  to  turn 
the  system  through  an  infinitely  small  angle,  and  let  B  and 
D  he  the  new  positions  of  A  and  C.  Draw  OE,  Ba,  and 
Dc  perpendicular  to  PE ;  draw  also,  AO,  BO,  CO,  and 
DO.  Denote  OA  by  r,  OC  by  r',  OE  by  p,  and  the  path 
described  by  a  point  at  a  unit's  distance  from  0,  by  6', 
Since  the  angles  A  OB,  and  COD  are  equal,  from  the  nature 
of  the  motion,  we  liave,  AB  =  H',  and  CD  =  r'  d' ;  and  since 
the  angular  displacement  is  infinitely  small,  these  may  be 
regarded  as  straight  lines  perpendicular  to  OA  and  OC. 
From  the  right-angled  triangles  ABa  and  CDc,  we  have, 

Aa  —  rd'cosBAa,  and  Co  =  r'G'cosDCc. 

In  the  right-angled  triangles  ABa,  and  OAE,  AB  is 
perpendicular  to  OA,  and  Aa  to  OE ;  hence,  BAa,  and 
AOE,  are  equal;  hence, 

P 
cosBAa  =  cos^  OE  —  - . 
r 

In  like  manner, 

cosDCc  =  cosCOEz=^,. 
r 

Substituting  in  the  preceding  equations,  we  have, 

Aa  =  p&',  and  Cc=]y^'j    -'.  Aa=Coj 
whence, 

P.Aa=  P.Cc=  Pp&'. 

The  first  member  of  the  equation  is  this  quantity  of  work 
of  P,  Avhen  its  point  of  application  is  A  ;  the  second  is  its 
quantity  of  work,  when  the  point  of  application  is  at  C. 
Hence,  we  conclude,  that  the  ele^nenfary  quantity  of  ivork 
of  a  rotating  force  is  always  the  same,  ivherever  its  point  of 
application  may  'be  tahen,  provided  its  line  of  direction  re- 
main unchanged. 

We  conclude,  also,  that  the  elementary  quantity  of  work 


186  MECHANICS. 

is  equal  to  the  intensity  of  the  force  multiplied  by  its  lever 
arm  into  the  elementary  space  described  by  a  point  at  a 
unit's  distance  from  the  axis. 

If  we  suppose  the  force  to  act  for  a  unit  of  time,  the 
intensity  and  lever  arm  remaining  the  same,  and  denote 
the  angular  velocity ,  by  4,  we  shall  have, 

Q'  =  Pp&. 

For  any  number  of  forces  similarly  applied,  we  shall 
have, 

Q  =  I{Pp)& (107) 

If  the  forces  are  in  equilibrium,  we  have,  (Art.  34), 
^(Pp)  =  0 ;  consequently,  Q  =  0. 

Hence,  if  any  number  of  forces  tending  to  produce 
rotation  are  in  equilibrium,  the  entire  quantity  of  work  of 
the  forces  is  equal  to  0. 

Accumulation  of  Work. 

139.  When  a  force  acts  on  a  body,  to  impart  motion,  it 
expends  a  certain  quantity  of  work  in  overcoming  the 
body's  inertia.  This  work  is  said  to  be  stored  up  in  the 
body;  and  if  a  resistance  be  offered  to  its  motion,  the  entire 
quantity  of  work  will  be  given  out,  and  expended  on  the 
resistance.  A  body  in  motion  may,  therefore,  be  regarded 
as  the  representative  of  a  quantity  of  work  which,  under 
certain  circumstances,  is  capable  of  being  utilized.  The 
work  stored  up,  or  accumulated,  depends  on  the  mass  of 
the  moving  body,  and  also  on  the  velocity  with  which  it 
moves.  To  find  an  expression  for  it,  let  us  denote  the 
weight  of  the  body  by  W,  its  velocity  by  v,  and  the 
quantity  of  accumulated  work  by  Q.  If  we  suppose  it  to 
be  projected  vertically  upward,  with  the  velocity,  v,  it  will 
rise  to  the  height  due  to  that  velocity,  that  is,  the  work 


CURVILINEAR   AND    ROTARY    MOTION.  187 

stored  up  in  the  body  is  sufficient  to  raise  the  weight,  Wy 
through  a  height,  h.     Hence, 

Q  =  Wh. 

But,  h  ~  |— ,  (Art.  105).      Substituting  this  value  of  A, 

we  have, 

W 

Denoting  the  mass  of  the  body  by  M,  we  have,  (Art.  15), 

W 

—  =  M,  and  this,  in  tlie  preceding  equation,  gives, 

Q  =  lMv\ 

Hence,  the  accumulated  ivork  in  a  moving  hody  is  equal  to 
one-half  the  tody's  mass  into  the  square  of  its  velocity. 

The  expression  ^Mv"^  is  called  the  living  force  of  the 
body.  Hence,  the  living  force  of  a  hody  is  equal  to  half  its 
mass,  inultipUed  ly  the  square  of  its  velocity.  The  living 
force  of  a  body  is  the  measure  of  the  quantity  of  work 
expended  in  producing  the  velocity,  or,  of  the  quantity  of 
work  the  body  is  capable  of  giving  out. 

When  forces  tend  to  increase  the  velocity,  their  work  is 
positive ;  when  they  tend  to  diminish  it,  their  work  is 
negative.  It  is  the  aggregate  of  all  the  work  expended, 
both  positive  and  negative,  that  is  measured  by  the  quan- 
tity, ^mv"^. 

If,  at  any  instant,  a  body  whose  mass  is  m,  has  a  velocity 
V,  and,  at  a  subsequent  instant,  its  velocity  has  become  v', 
we  have  for  the  accumulated  work  at  these  two  instants 

Q  =  ^7nv^,     Q'  =  ^7nv'^; 

and,  for  the  aggregate  quantity  of  work  expended  in  the 
interval, 

Q"  ^^iy'-'-v-") (108) 


188  MECHANICS. 

When  tlie  motive  forces,  during  the  interval,  perform 
more  work  than  the  resistances,  v'  is  greater  than  v,  and 
there  is  an  accumulation  of  work.  When  .the  work  of  the 
resistances  exceeds  that  of  the  motive  forces,  v  exceeds  i'', 
Q",  is  negative,  and  there  is  a  loss  of  work  which  is  ex- 
pended on  the  resistances. 

Living  Force  of  Revolving   Bodies. 

140.  Denote  the  angular  velocity  of  a  revolving  body  by 
&,  the  masses  of  its  elementary  particles  by^«,  7n',  &c.,  and 
their  distances  from  the  axis  of  rotation,  by  r,  r,  &c.  Their 
velocities  will  be  r&,  r'&,  &c.,  and  their  living  forces,  ^mr^^^, 
^m'r"^^^,  &c.  Denoting  the  entire  living  force  of  the  body 
by  X,  we  have,  by  summation,  recollecting  that  &  is  the 
same  for  all  the  terms, 

L=^I{mry (109) 

But  I{m?'^)  is  the  moment  of  inertia  of  the  body  with 
respect  to  the  axis  of  rotation.  Denoting  the  entire  mass 
by  M,  and  its  radius  of  gyration,  with  respect  to  the  axis  of 
rotation,  by  k,  we  have, 

I(mr')  =  Mk';     :.    L  =  ^^MFd' (110) 

If,  at  any  subsequent  instant,  the  angular  velocity  has 
become  &',  we  have, 

L'  ^  iMk'r; 

and,  for  the  gain  or  loss  of  living  force  in  the  interval, 

L"  =  iMk'{r-^') (Ill) 

If,  in  equation  (110),  we  make  ^  =  1,  we  have, 

L'"  =  i^(mr') ;  or,  2'(mr')  =  2Z. 

That  is,  the  moment  of  inertia  of  a  body,  with  respect  to 
an  axis,  is  equal  to  twice  its  living  force  when  the  angular 


CURVILINEAE    AND    EOTAKY    MOTION.  189 

velocity  is  equal  to  1,  or,  to  twice  the  quantity  of  work  that 
must  be  expended  to  generate  a  unit  of  angular  velocity. 

The  principle  of  living  force  is  applied  in  discussing  the 
motion  of  machines.  When  the  power  performs  more  work 
than  is  necessary  to  overcome  the  resistances,  the  velocities 
of  the  parts  increase,  and  a  quantity  of  Avork  is  stored  up, 
to  he  given  out  again  when  the  resistances  require  more 
work  to  overcome  them  than  is  furnished  by  the  motor. 

In  many  machines,  pieces  are  introduced  to  equalize  the 
motion  ;  this  is  particularly  the  case  when  either  the  power 
or  the  resistance  is  variable.  Such  pieces  are  called /y- 
luheels. 

Fly-Wheels. 

141.  A  fly-wheel  is  a  heavy  wheel  mounted  on  an  axis, 
near  the  point  of  application  of  the  force  it  is  designed  to 
regulate.  It  is  generally  composed 
of  a  rim,  connected  with  the  axis  by 
radial  arms.  Sometimes  it  consists 
of  radial  bars,  carrying  spheres  of 
metal  at  their  outer  extremities. 
Let  us  denote  the  mass  of  the  wheel 
by  M,  its  radius  of  gyration  by  k,  the 
quantity  of  work  stored  up   in  any  ^^^'  ^^^' 

time  by  §,  and  the  niitial  and  terminal  angular  velocities 
by  &'  and  &".     We  shall  have,  from  equation  (111), 
Q  =  iMk\d"' -  &") (112) 

If  ^">4',  Q  is  positive  and  work  is  stored  up  ;  if,  ^" <&^, 
Q  is  negative,  and  the  wheel  gives  out  work. 

If  the  angular  velocity  increase  from  &'  to  ^",  and  then 
decrease  to  &',  and  so  on,  alternately,  the  work  accumulated 
during  the  first  part  of  each  cycle  is  given  out  during  the 
second  part,   and  any  device  that  will  make  &'  and  d"  more 


190  MECHANICS. 

nearly  equal,  will  contribute  toward  equalizing  the  motion 
of  the  wheel.  By  suitably  increasing  the  mass  and  radius 
of  gyration,  their  difference  may  be  made  as  small  as  de- 
sirable. Let  the  half-sum  of  the  greatest  and  least  angular 
velocities  be  called  the  mean  angular  velocity,  and  denote 

it  by  S"'.     We  shall  have  — —  =  d",  and  by  factoring  the 

second  member  of  (lli^),  we  have, 

whence,  by  substituting  the  value  of  d"  +  a', 

Q  =  Mk\d"  -  ^')a  '' (113) 

Let  us  suppose  the  difference  between  the  greatest  and 
least  velocity,  equal  to  the  n^^  part  of  their  mean,  that  is, 
that 


This,  in  (113),  gives 

From  this  equation  the  moment  of  inertia  of  the  wheel 
may  be  found,  when  we  know  m,  Q,  and  &'".  The  value  of 
n  may  be  assumed ;  for  most  kinds  of  work  a  value  of 
from  6  to  10  will  be  found  to  give  sufficient  uniformity ; 
the  value  of  &'"  depends  on  the  character  of  the  work  to  be 
performed,  and  Q  is  made  known  by  the  character  of  the 
motion  to  be  regulated. 


Composition  of  Rotations. 
142.  Let  a  body,  ACBD,  be  acted  on  by  an  impulse  that 
would  cause  it  to  revolve  about  AB  with  an  angular  velo- 
city V.  and  at  the  same  instant  let  it  be  acted  on  by  a  second 
impulse  that  would  cause  it  to  revolve  about  DC  with  an 


CUBVILINEAR    AKD    ROTAKY    MOTIOJS^.  191 

angular  velocity  v'.  Suppose  the  axes  to  intersect  at  0,  and 
from  any  assumed  point  in  tlieir  plane,  draw  perpendicu- 
lars to  AB  and  DC,  denoting  the  former  by  x  and  the 
Utter  by  y.  Then  will  the  velocity  of  the  assumed  point 
due  to  the    first  force  be  vx,  and 

its  velocity  due  to  the  second  force  ^ -^ 

will  be  v'y.      Now,  the  point  can  /^  /  N. 

always  be  taken,  so  that  rotation       /  a/    ~'^^fK\ 

about  the  first  axis  shall  tend  to      [  /\^>^/       | 

depress  the  point  below  the  plane,    A^^  ^  ^ -h 

and  about  the  second  axis  to  elevate       V  / 

it  above  the  plane.      In  this   case  \ ^^/ 

the  effective  velocity  of  the  point  is  j,.^ 

v'y  —  vx. 

If  this  velocity  is  equal  to  0,  the  assumed  point  remains 
fast,  and,  we  have, 

vx  ~v'y ;  OY,  X  :  y  :  :  v'  :  v (114) 

To  find  the  position  of  the  point,  in  the  case  supposed, 
lay  off  OH  equal  to  v,  and  01  equal  to  v',  and  on  these  as 
sides,  construct  the  parallelogram  IH,  and  draw  its  diagonal 
OK,  Then  will  any  point,  F,  of  this  diagonal  satisfy  pro- 
portion (114).  For,  let  OH  and  01  for  a  moment  be 
regarded  as  forces,  and  OK  their  resultant,  and  suppose  PF 
and  FG  to  be  perpendicular  to  OH  and  01.  Then  if  P 
be  taken  as  a  centre  of  moments,  we  have,  (Art.  34), 
OHx  FF=  01 X  FG;  or,  ?;  X  FF=  v'  X  FG. 

From  which  we  find, 

FF  '.  FG  \\  V   \  v;  or,  FF  -.  FG  w  x  \  y. 

Hence,  every  point  of  OK  remains  at  rest ;  it  is  conse- 
quently tlie  resultant  axis  of  rotation.  We  have,  therefore, 
the  following  principles: 

If  a  body  be  acted  on  simultaneously  by  two  impulses,  each 


192  MECHANICS. 

tending  to  impart  rotation  about  a  separate  axis,  the  result- 
ant motion  will  be  one  of  rotation  ahout  a  third  axis  lying 
in  the  plane  of  the  other  ttvo,  and  passing  through  their 
p)oint  of  intersection. 

Tlie  direction  of  the  resultant  axis  coincides  with  the 
diagonal  of  a  parallelogram,  whose  sides  are  the  component 
axes,  and  whose  lengths  are  proportional  to  the  angular  ve- 
locities. 

Let  (9// and  0/ represent  the  angular  velocities  v  and  v', 
and  OK  the  diagonal  of  the  parallelogram  constructed  on 
these  lines  as  sides.     Take  any  point,  /,  on  ^ 

the  second  axis,  and  let  fall  perpendiculars 
on  O^and  OK;  denote  the  former  by  r ,  and 
the  latter  by  r" ;  denote,  also,  the  angular   o       '     H 
velocity  about  OK,  by  v".     Since  the  space  ^^=-  ^^^ 

passed  over  by  /,  in  any  time,  t,  depends  only  on  the  first 
force,  it  will  be  the  same  whether  we  regard  the  revolution 
as  taking  place  about  OH  or  OK.  If  Ave  sujipose  the 
rotation  to  take  place  about  OH,  the  space  passed  over  in 
the  time,  t,  will  be  rvt ;  if  we  suppose  the  rotation  to  take 
place  about  OK,  the  space  passed  over  in  the  same  time 
will  be  r''v"t.  Placing  these  equal,  we  have,  after  reduc- 
tion, 

""=^,^' (115) 

If  we  suppose,  as  before,  that  OH  and  01  are  forces,  and 
)K  their  resultant,  and  take  /  as  a  centre  of  moments, 

we  have, 

r 

OK  X  r"  =  vr ;   or,   OK  =  —v. 

r 

By  comparing  this  with  equation  (115),  we  have, 

v"  =  OK. 

Hence,  the  resultant   aiigular  velocity  is  equal  to  the 


CURVILINEAR    AJSTD    ROTARY    MOTIOIf.  193 

diagonal  of  the  parallelogram  described  on  the  component 
angular  velocities  as  sides. 

By  a  course  of  reasoning  similar  to  that  employed  in 
demonstrating  the  parallelopipedon  of  forces,  we  might 
show,  that, 

If  a  body  be  acted  on  by  three  simultaneous  impulses, 
each  tending  to  2^'^oduce  rotation  about  axes  intersecting, 
the  resultant  motion  will  be  one  of  rotation  about  the  diag- 
onal of  the  imrallelopipedon  ivhose  adjacent  edges  are  the 
component  angular  velocities,  and  the  resultant  angular 
velocity  will  be  equal  to  the  length  of  this  diagonal. 

The  principles  just  deduced  are  called  the  parallelogram 
dnd  the  parallelopipedon  of  rotations. 

Application  to  the  Gyroscope. 

143.  The  gyroscope  is  an  instrument  that  may  be  used 
to  illustrate  the  laws  of  rotary  motion.  It  consists  of  a 
heavy  wheel,  A,  mounted 
on  an  axle,  BC.  This  y'X- 
axle  is  attached,  by  pivots,  '^ 
to  the  inner  edge  of  a  cir- 
cular hoop,  DE,  within 
wiiich   the  wheel.  A,   can  ^^^-  ^^'^• 

turn  freely.  On  one  side  of  the  hoop,  and  in  the  prolongation 
of  the  axle,  BC,  is  a  bar,  EF,  having  a  conical  hole  drilled 
on  its  lower  face  to  receive  the  point  of  a  vertical  standard, 
O.  If  a  string  be  wrapped  round  the  axle,  BC,  and  then 
rapidly  unwound,  so  as  to  impart  a  motion  of  rotation  to 
the  wheel.  A,  in  the  direction  indicated  by  the  arrow-head, 
it  is  observed  that  the  machine,  instead  of  sinking  down- 
ward under  the  action  of  gravity,  takes  up  a  retrograde 
orbital  motion  about  G,  as  indicated  by  the  arrow-head,  H. 
For  a  time,  the  orbital  motion  increases,  and,  under  certain 

9 


194  MECHANICS. 

oircamstances,  the  bar,  EF,  is  observed  to  rise  upward  in 
a  retrograde  spiral  direction ;  and,  if  the  cavity  for  receiv- 
ing the  pivot  is  pretty  shallow,  the  bar  may  even  be  thrown 
utf  the  standard.  Instead  of  a  bar,  EF,  the  instrument  may 
simply  have  an  ear  at  E,  and  be  suspended  by  a  string. 
The  phenomena  are  the  same  as  before. 

Before  explaining  the  phenomena,  it  will  be  necessary  to 
assume  conventional  rules  for  giving  signs  to  the  different 
rotations. 

Let  OX,  0  Y,  and  OZ,  be  three  rectangular  axes.     It  has 
been  agreed  to  call  all  distances,  estimated  from  0,  toward 
either  X,  F,  or  Z,  positive ;  con- 
sequently,  all  distances  estimated 
in  contrary  directions  must  be  n       )         D'il 

negative.      If    a     body    revolve        J^  ~~^ 


about  either  axis,  in  such  a  man-     y"     ^  \J 

ner  as  to  appear  to  an  eye  on  the  Fig.  iis. 

positive  portion  of  the  axis,  and  looking  toward  the  origin, 
to  move  in  the  same  direction  as  the  hands  of  a  watch,  that 
rotation  is  called  positive.  If  rotation  take  place  in  an 
opposite  direction,  it  is  negative.  The  arrow-head.  A,  indi- 
cates the  direction  of  positive  rotation  about  the  axis  of  X, 
the  arrow-head,  B,  the  direction  of  positive  rotation  about 
the  axis  of  Y,  and  the  arrow-head,  C,  the  direction  of  posi- 
tive rotation  about  the  axis  of  Z. 

Suppose  the  axis  of  the  wheel  of  the  gyroscope  to 
coincide  with  the  axis  of  X,  taken  horizontal ;  let  the 
standard  coincide  with  the  axis  of  Z,  the  axis  of  Y  being 
perpendicular  to  both.  Let  positive  rotation  be  communi- 
cated to  the  wheel  by  a  string,  and  then  let  the  instru- 
ment be  abandoned  to  the  action  of  gravity.  During  the 
first  instant,  the  force  of  gravity  will  impart  to  it  a  positive 
rotation  about  the  axis  of  Y,    Denote  the  angular  velocity 


CURVILINEAR    AND    ROTARY    MOTION.  195 

about  the  axis  of  JT,  by  v,  and  about  the  axis  of  Y,  by  v'; 
lay  off  in  a  positive  direction  on  the  axis  of  JT,  OB  equal  to 
V,  and  on  the  positive  direction  of  the  axis  of  Y,  OP  equal 
to  v',  and  complete  the  parallelogram,  OF.  Then  will 
OF  represent  the  resultant  axis  of  revolution,  and  the 
angular  velocity.  In  moving  from  OD  to  OF,  the  axis 
has  a  positive,  or  retrograde  orbital  motion  about  the  axis 
of  Z.  To  construct  the  resultant  axis  for  the  second 
instant,  we  must  compound  three  angular  velocities.  Lay 
off  on  a  perpendicular  to  OF  and  OZ,  the  angular  velocity 
due  to  gravity,  and  on  OZ  the  angular  velocity  in  the 
orbit ;  construct  a  parallel opipedon  on  the  three  velocities, 
and  draw  its  diagonal  through  0.  This  diagonal  will 
coincide  with  the  axis  for  tne  second  instant,  and  will 
represent  the  resultant  angular  velocity.  For  the  next 
instant,  we  proceed  as  before,  and  so  on  continually.  Since, 
in  each  case,  the  diagonal  is  greater  than  either  edge  of  the 
parallelopipedon,  it  follows  that  the  angular  velocity  will 
continually  increase,  and,  were  there  no  hurtful  resistance, 
this  increase  would  go  on  indefinitely.  The  effect  of  gravity 
is  continually  exerted  to  depress  the  centre  of  gravity 
of  the  instrument,  whilst  the  effect  of  the  orbital  rotation 
is  to  elevate  it.  When  the  latter  prevails,  the  axis  of  the 
gyroscope  rises;  when  the  former  prevails,  the  gyroscope 
descends.  Whether  one  or  the  other  of  these  conditions 
is  fulfilled,  depends  on  the  angular  velocity  of  the  wheel, 
and  the  position  of  the  centre  of  gravity  of  the  instrument. 
Were  the  instrument  counterpoised  so  as  to  place  the  centre 
of  gravity  exactly  over  the  pivot,  there  would  be  no  orbital 
motion,  neither  would  the  instrument  rise  or  fall.  Were 
the  centre  of  gravity  thrown  on  the  opposite  side  of  the 
pivot,  the  rotation  due  to  gravity  would  be  negative,  and 
the  orbital  motion  would  be  direct. 


CHAPTER  VII. 

MECHANICS    OF    LIQUIDS. 

Classification  of  Fluids. 

144.  A  FLUID  is  a  body  whose  particles  move  freely 
amongst  each  other,  each  particle  yielding  to  the  slightest 
force. 

Fluids  are  of  two  classes :  liquids,  of  which  water  is  a 
type,  and  gases,  or  vapors,  of  which  air  and  steam  are  types. 
The  distinctive  property  of  the  first  class  is,  that  they  are 
almost  incompressible ;  thus,  water,  on  being  pressed  by  a 
force  of  15  lbs.  on  each  square  incli  of  surface,  suffers  a 
diminution  of  not  more  than  the  g-ooVoirth  of  its  bulk. 
Bodies  of  the  second  class  are  readily  compressible;  thus, 
air  and  steam  are  easily  compressed  into  smaller  volumes, 
and  when  the  pressure  is  removed,  they  expand  and  occupy 
larger  volumes. 

Most  liquids  are  imperfect;  that  is,  there  is  more  or  less 
adherence  between  their  particles,  giving  rise  to  viscosity. 
In  what  follows,  they  will  be  regarded  as  destitute  of  vis- 
cosity, and  homogeneous.  In  certain  cases  fluids  may  also 
be  regarded  as  destitute  of  weight,  without  impairing  the 
validity  of  the  conclusions. 

Principle  of  Equal   Pressures. 

145.  From  the  nature  of  a  fluid,  each  of  its  particles  is 
perfectly  movable  in  all  directions.  From  this  we  deduce 
the  following  fundamental  law,  viz. :  If  a  fluid  he  in  equi- 
librium, under  the  action  of  any  forces  whatever,  each  ptar- 


i^-"^^ 


MECHANICS   OF   LIQUIDS.  197 

tide  of  the  mass  is  equally  pressed  in  all  directions  ;  for,  if 
any  particle  were  more  strongly  pressed  in  one  direction 
than  in  the  others,  it  would  yield  in  that  direction,  and 
motion  would  ensue,  which  is  contrary  to  the  hypothesis. 

This  is  called  the  'principle  of  equal  pressures. 

It  follows  from  the  principle  of  equal  pressures,  that  if  a 
fluid,  confined  in  a  vessel,  be  pressed  at  any  part  of  its  sur- 
face, the  pressure  will  be  transmitted  without  change  of 
intensity  to  every  part  of  the  inner  surface  of  the  vessel. 

This  may  be  illustrated  as  follows:  let  a  vessel,  AB,  be 
filled  with  water,  and  let  two  pistons,  C  and  D,  be  fitted  to 
corresponding  openings  in  the  side  of 
the  vessel,  and  suppose  the  fluid  to  be 
in  equilibrium.  If  any  extraneous 
force   be   applied   to   either  piston,  a        /  \ 

second  force  must  be  applied  to  the     M  JB 

other  to  hold  the  first  in  equilibrium,        \  / 

and  it  will  be  found  that  these  forces  ^^ — ^ 

are   proportional   to  the  areas  of  the  ^'^'  ^^^* 

pistons  to  which  they  are  applied.  This  relation  holds  true, 
no  matter  what  the  areas  of  the  pistons,  or  at  what  portion 
of  the  vessel  they  may  be  applied. 

A  pressure  transmitted  through  a  fluid  in  equilibrium, 
to  the  surface  of  a  containing  vessel,  is  normal  to  that  sur- 
face; for  if  it  were  not,  we  might  resolve  it  into  two 
components,  one  normal  to  the  surface,  and  the  other  tan- 
gential ;  the  effect  of  the  former  would  be  destroyed  by  the 
resistance  of  the  vessel,  whilst  the  latter  would  impart 
motion  to  the  fluid,  which  is  contrary  to  the  supposition 
of  equilibrium.  In  like  manner,  it  may  be  shown,  that 
the  resultant  of  all  the  pressures,  acting  at  any  point  of 
the  free  surface  of  a  fluid,  is  normal  to  the  surface  at  that 
point. 


198  MECHANTCS. 

When  the  only  force  acting  is  gravity,  the  surface  is  level. 
For  small  areas,  a  level  surface  coincides  sensibly  with  a 
horizontal  plane.  For  larger  areas,  as  lakes  and  oceans,  a 
level  surface  coincides  with  the  general  surface  of  the  earth. 
AVere  the  earth  at  rest,  the  level  surface  of  lakes  and  oceans 
would  be  spherical ;  but,  on  account  of  the  centrifugal  force 
arising  from  the  rotation  of  the  earth,  it  is  that  of  an  ellip- 
soid, whose  axis  of  revolution  is  the  axis  of  the  earth. 

Pressure  due  to   Weight, 

146.  If  an  incompressible  fluid  be  in  equilibrium,  the 
pressure  at  any  point  arising  from  the  weight  of  the  fluid, 
is  proportional  to  the  depth  of  that  point  below  the  free 
surface. 

Take  an  infinitely  small  surface,  supposed  horizontal,  and 
conceive  it  to  be  the  base  of  a  vertical  prism  whose  altitude 
is  its  distance  from  the  free  surface.  Let  this  filament  be 
divided,  by  horizontal  planes,  into  infinitely  small,  or  ele- 
mentary prisms.  From  the  principle  of  equal  pressures, 
the  pressure  on  the  lower  face  of  any  one  of  these  prisms 
is  greater  than  that  on  its  upper  face,  by  the  weight  of  the 
prism,  whilst  the  lateral  pressures  counteract  each  other. 
Hence,  the  pressure  on  the  lower  face  of  the  first  prism 
from  the  top,  is  equal  to  its  weight ;  that  on  the  lower  face 
of  the  second  is  equal  to  the  weight  of  the  first,  plus  the 
weight  of  the  second,  and  so  on  to  the  bottom.  Hence, 
the  pressure  on  the  assumed  surface  is  equal  to  the  weight 
of  the  entire  column  of  fluid  above  it.  Had  the  assumed 
elementary  surface  been  oblique  to  the  horizon,  or  perpen- 
dicular to  it,  and  at  the  same  depth  as  before,  the  pressure 
on  it  would  have  been  the  same,  but  its  direction  would 
have  been  normal  to  the  surface.  We  have,  therefore,  the 
following  law : 


MECHANICS   OF  LIQUIDS.  199 

Tlie  pressure  on  an  elementary  portion  of  the  surface  of  a 
vessel  containing  a  heavy  fluid,  is  equal  to  the  iceight  of  a 
prism  of  the  fluid,  ivhose  lase  is  the  surface  pressed,  and 
whose  altitude  is  its  depth  below  the  free  surface  of  the  fluid. 

Denoting  the  area  of  the  elementary  surface,  by  5,  its 

depth  below  the  free  surface,  by  z,  the  weight  of  a  unit  of 

volume  of  the  fluid,  by  w,  and  the  pressure,  by  p,  we  shall 

have, 

p  =  wzs (116) 

We  have  seen  that  the  pressure  on  any  element  of  a  sur- 
face is  normal  to  the  surface.     Denote  the  angle  this  nor- 
mal makes  with  the  vertical,  estimated  from 
above  downward,  by  (p,  and  resolve  the  pres-         . /I    (0/ 
sure  into  two  components,  one  vertical  and      gj^  F% 
the  other  horizontal ;  denoting  the  vertical    /___/ 
component  hj])',  we  have.  Fig.  120. 

2/  ~  wzscos:p (ll*^) 

But  scoscp  is  equal  to  the  horizontal  projection  of  the 
element  s,  in  other  words,  it  is  a  horizontal  section  of  a 
vertical  prism,  of  which  that  surface  is  the  base. 

Hence,  the  vertical  component  of  the  pressure  on  any 
element  of  the  surface  is  equal  to  the  weight  of  a  column  of 
the  fluid,  whose  base  is  the  horizontal  projection  of  the 
element,  a7id  whose  altitude  is  the  distance  of  the  element 
from  the  free  surface  of  the  fluid. 

The  distance,  z,  has  been  taken  dispositive  from  the  sur- 
face of  the  fluid  downward.  If  9  <  90°,  we  have  coscp  posi- 
tive ;  hence  p',  will  be  positive,  which  shows  that  the  ver- 
tical pressure  is  exerted  downward.  If  <p>90°,  we  have 
cosip  negative,  hence  p'  is  negative,  which  shows  that  the 
vertical  pressure  is  exerted  upward  (see  Fig.  120). 

Suppose  the  interior  surface  of  a  vessel  containing  a 


200  MECHANICS. 

heavy  fluid  to  be  divided  into  elementary  portions,  whose 
areas  are  denoted  by  s,  s',  s",  &c.;  denote  the  distances  of 
these  elements  from  the  free  surface,  by  z,  z',  z",  &c.  From 
the  principle  just  demonstrated,  the  pressures  on  these  sur- 
faces will  be  tusz,  ws'z\  ivs"z",  &c.,  and  the  entire  pressure 
on  the  interior  of  the  vessel  will  be  equal  to, 

w{sz  +  s'z'  +  s"z"  +  &c.) ;  or,  w  X  ^sz). 

Let  Z  denote  the  depth  of  a  column  of  fluid,  whose  base 
is  the  surface  pressed,  and  whose  weight  is  equal  to  the 
entire  pressure,  then  will  this  pressure  be  2v{s  -i-  s'  +  s" 
+  &c.)Z ;  or,  ivZ .  Is.     Equating  these  values,  we  have, 

to  .  I(sz)  =  toZ .  I(s),     .-.     Z=  -^ (118) 

The  second  member  of  (118)  is  the  distance  of  the  cen- 
tre of  gravity  of  the  surface  pressed,  from  the  free  surface 
of  the  fluid.     Hence, 

The  pressure  of  a  heavy  fluid  on  the  interior  of  a  vessel 
is  equal  to  the  iceight  of  a  cylinder  of  the  fluid,  tvhose  base 
is  the  area  pressed,  and  luhose  altitude  is  the  distance  of  its 
centre  of  gravity  from  the  free  surface  of  the  fluid. 

Examples. 

1.  A  hollow  sphere  is  filled  with  a  liquid-  How  does  the  pressuie, 
on  the  interior  surface,  compare  with  the  weight  of  the  liquid  ? 

SOLUTION. 

Denote  the  radius  of  the  surface,  by  r,  and  the  weight  of  a  unit  of 
the  liquid,  by  w.    The  surface  pressed  is  ^nr^ ;  and,  its  centre  of 
gravity  is  at  a  distance  r  from  the  free  surface  of  the  liquid ;  thence 
the  pressure  on  the  interior  surface  is  equal  to, 
w  X  Atij-^  X  r  =  Anwi'^. 

But  the  weight  of  the  liquid  is  equal  to 

That  is,  the  entire  pressure  is  three  times  the  weight  of  the  liquid. 


MECHANICS   OF   LIQUIDS.  201 

2.  A  hollow  cylinder,  with  a  circular  base,  is  filled  with  a  liquid. 
How  does  the  pressure  on  the  interior  surface  compare  with  the 
weight  ot  the  liquid  ? 

SOLUTION. 

Denote  the  radius  of  the  base,  by  r,  and  the  altitude,  by  /t.  The 
centre  of  gravity  of  the  lateral  surface  is  at  a  distance  from  the  upper 
surface  of  the  fluid  equal  to  i/i.  If  we  denote  the  weight  of  the  unit 
of  volume  of  the  liquid,  by  w,  we  have,  for  the  pressure  on  the  interior 
surface, 

wJiTcr'^  -\-  2w7tr  .  ^i"^  =  W7trh{r  +  ^0- 

But  the  weight  of  the  liquid  is  equal  to 

r  4-  a 
Hence,  the  total  pressure  is  — —  times  tJie  weight  of  tJie  liquid. 

If  h  —  r,  the  pressure  is  twice  the  weight. 

If  r  =  2/i,  the  pressure  is  f  of  the  weight. 

If  A,  =  2r,  the  pressure  is  three  times  the  weight,  and  so  on. 

In  all  cases,  the  pressure  exceeds  the  weight  of  the  liquid. 

3.  A  right  cone,  with  a  circular  base,  stands  on  its  base,  and  is 
filled  with  a  liquid.  How  does  the  pressure  on  the  internal  surface 
compare  with  the  weight  of  the  liquid  ? 

SOLUTION. 

Denote  the  radius  of  the  base,  by  r,  and  the  altitude,  by  7i,  then 
will  the  slant  height  be  equal  to 

y/h^  +  r\ 
The  distance  of  the  centre  of  gravity  of  the  lateral  surface,  below 
the  free  surface  of  the  liquid,  is  p.    If  we  denote  the  weight  of  a  unit 
of  volume  of  the  liquid,  by  w,  we  have,  for  the  total  pressure  on  the 
interior  surface. 


WTtrVi  +  IwTtrh  s/h'  +  r'  =  WTtrh{r  +  %  \/A^  -f  ?•='). 
But  the  weight  of  the  liquid  is 

^witr'^h  =  WTtrh  X  ^r. 

Hence,  the  total  pressure  is  equal  to times  the  weight. 

4.  Required  the  relation  between  the  pressure  and  the  weight  in 
the  preceding  case,  when  the  cone  stands  on  its  vertex. 

9* 


202  MECHANICS. 


SOLUTION. 

The  total  pressure  is 


and,  consequently,  it  is  equal  to "*"      times  the  weight  of  the  liquid. 

r 

5.  What  is  the  pressure  on  the  lateral  faces  of  a  cubical  vessel  filled 
with  water,  the  edge  of  the  cube  being  4  feet,  and  the  weight  of  the 
water  63^  lbs.  per  cubic  foot  ?  Ans.  8000  lbs. 

6.  A  cylindrical  vessel  is  filled  with  water.  The  height  of  the 
vessel  is  4  feet,  and  the  radius  of  the  base  6  feet.  What  is  the  pres- 
sure on  the  lateral  surface  ?  Ans    18850  lbs.,  nearly. 


Centre  of  Pressure  on  a  Plane  Surface. 

147.  The  centre  of  pressure  on  a  surface,  is  the  point 
at  which  the  resultant  pressure  intersects  the  surface. 

Let  A  BCD  be  a  plane,  pressed  by  a  liuid  on  its  upper 
surface,  AB  its  intersection  with  the  free  surface  of  the 
fluid,  G  its  centre  of  gravity,  0  the       ^  ^ 

centre  of  pressure,  and  s  the  area  of     j- — ./S^^.^^ 

an  element  of  the  surface  at  S.     De-  "^"-\/    i)s   / 

note  the  inclination   of  the  plane  to  /^^/ 

the  horizontal,  by  a,  the  distances  from  ^5.,^     / 

0  to  AB,  by  X,  from  O  to  AB,  by  ;;,  ^^ 

and  from  S  to  AB,  by  r.      Denote,  ^'^'  ^'^^^ 

also,  the  area  A  C,  by  A,  and  the  weight  of  a  unit  of  volume 
of  the  fluid,  by  w.  The  distance  from  G  to  the  free  sur- 
face of  the  fluid,  is  ;;  sina,  and  that  of  any  element  of  the 
plane,  is  r  sina. 

From  the  preceding  article,  we  see  that  the  entire  pres- 
sure is  ?6'^4^sina,  and  its  moment,  with  respect  to  AB,  is, 

wA2)Bma.  X  X. 

The  elementary  pressure  on  s,  in  like  manner,  is  7csr  sina. 


MECHANICS    OF    LIQUIDS.  203 

its  moment,  with  respect  to  AB,  is  tusr"^  sina,  and  the  sum 
of  all  the  elementary  moments  is, 
w^ivLOLS{sr'^). 

But  the  resultant  moment  is  equal  to  the  algebraical 
sum  of  the  elementary  moments.     Hence, 

w^j;sina  X  x  =  w  sina  2'(5r') ; 

and,  by  reduction. 

The  numerator,  is  the  moment  of  inertia  oiABCD,  with 
respect  to  AB,  and  the  denominator  is  the  moment  of  the 
area  with  respect  to  the  same  line.  Hence,  the  distance 
from  the  centre  of  pressure  to  the  intersection  of  the  plaiie 
with  the  surface,  is  equal  to  the  moment  of  inertia  of  the 
pla7ie,  divided  hy  the  moment  of  the  jilane. 

If  we  take  AD,  perpendicular  to  AB,  as  an  axis  of 
moments,  denoting  the  distance  of  0  from  it,  by  y,  and  of 
S  from  it,  by  I,  we  have, 

toAp  sinay  =:  toEiYLal^srl) ; 
and,  by  reduction, 

^=^ (-) 

The  values  of  x  and  y  determine  the  centre  of  pressure. 

It  may  be  observed  that  x  is  the  distance  from  AB  io 
the  centre  of  percussion  of  the  plane,  and  y  is  the  distance 
from  AD  to  the  centre  of  gravity  of  the  jolane.  Hence, 
the  centre  of  pressure  is  the  same  as  the  centre  of  per- 
cussion. 

Examples. 

1.  Where  is  the  centre  of  pressure  on  a  rectangular  flood-gate,  the 
upper  line  of  the  gate  coinciding  with  the  surface  of  the  water  ? 


^04  MECHANICS. 


SOLUTION. 

It  will  be  on  the  line  joining  the  middle  points  of  the  upper  and 
lower  edges  of  the  gate.  Denote  its  distance  from  the  upper  edge, 
by  2,  the  depth  of  the  gate,  by  21,  and  its  mass,  by  M.  The  distance 
of  the  centre  of  gravity  from  the  upper  edge  is  I. 

From  example  1,  (Art,  123),  we  have,  for  the  moment  of  inertia  of 
the  rectangle, 


i¥(^  +  p)  =  .¥ii«. 


But  the  moment  of  the  rectangle  is, 

Ml; 
hence,  by  division,  we  have, 

2  =  ^^  =  K20. 

That  is,  the  centre  of  pressure  is  two-thirds  of  the  distance  from  the 
upper  to  the  lower  edge  of  the  gate. 

2.  Let  it  be  required  to  find  the  pressure  on  a  sub-      e    (j    r 
merged  gate,  ABCB,  the  plane  of  the  gate  being  verti- 
cal ;  also,  the  distance  of  the  centre  of  pressure  below 


the  surface  of  the  water.  j^ 

SOLUTION. 


:C' 


B 


Let  EFhe  the  intei-section  of  the  plane  with  the  sur-      jy  q 

face  of  the  water,  and  suppose  the  rectangle,  AC,  to  be       p-^„  ^^2 
prolonged  till  it  reaches  EF.    Let  (7,  C\  and  6^",  be  the 
centres  of  pressure  of  the  rectangles  EC,  EB,  and  AC  respectively. 
Denote  GC",  by  z,  ED,  by  a,  and  EA,  by  a'.     Denote  the  breadth 
of  the  gate,  by  h,  and  the  weight  of  a  unit  of  volume  of  the  water, 
by  IB. 

The  pressure  on  EC  will  be  \a^hw,  and  the  pressure  on  EB  will  be 
\a"^hw ;  hence,  the  pressure  on  AC yv'iW  be, 

which  is  the  pressure  required ;  from  the  principle  of  moments,  the 
moment  of  the  pressure  on  AC,  is  equal  to  the  moment  of  the  pres- 
sure on  EC,  minus  the  moment  of  the  pressure  on  EB.     Hence, 


'  a"  -  (r 
which  is  the  required  distance  from  the  surface  of  the  water. 


MECHANICS    OF    LIQUIDS.  205 

3.  To  find  the  pressure  on  a  gate,  when  both  sides  are  pressed,  the 
water  being  at  different  levels  on  the  sides.  Also,  to  find  the  centre 
of  pressure. 


SOLUTION. 


Denote  the  depth  of  water  on  one  side,  by  a, 
and  on  the  other,  by  «',  the  other  elements  be- 
ing the  same  as  before. 

The  total  pressure  will  be,  — ^ 


Fig.  123. 


Estimating  z  from  C  upward. 


4.  A  sluice-gate,  10  feet  square,  is  placed  vertically,  its  upper  edge 
coinciding  with  the  surface  of  the  water.  What  is  the  pressure  on 
the  upper  and  lower  halves  of  the  gate,  respectively,  the  weight  of  a 
cubic  foot  of  water  being  C2i  lbs.  ? 

Am.  7812.5  lbs.,  and  23437.5  lbs. 

5.  What  must  be  the  thickness  of  a  rectangular'  dam  of  granite, 
that  it  may  neither  rotate  about  its  outer  edge,  nor  slide  along  its 
base,  the  weight  of  a  cubic  foot  of  granite  being  160  lbs.,  and  the 
coefficient  of  friction  between  it  and  the  soil  being  .G  ? 

SOLUTION. 

First,  to  prevent  rotation.  Denote  the  height  of  the  wall,  by  ^, 
and  suppose  the  water  to  extend  from  bottom  to  top.  Denote  the 
thickness,  by  t,  and  the  length,  by  I.     The  weight  in  pounds,  will  be, 

Ihi  X  160 ; 
and  this  being  exerted  through  its  centre  of  gravity,  the  moment  of 
the  weight  with  respect  to  the  outer  edge,  is, 

\mi  X  160  =  miw. 

The  pressure  of  the  water  against  the  inner  face,  in  pounds,  is 
equal  to 

\ni^  X  62.5  =:  IW  X  31. 2o. 
This  pressure  is  applied  at  the  centre  of  pressure,  which  is  (exam- 
ple 1)  at  a  distance  from  the  bottom  of  the  wall  equal  to  fjA;  hence, 
its  moment  with  respect  to  the  outer  edge  of  the  wall,  is  equal  to 
IK'  X  10.4166. 
The  pressure  of  the  water  tends  to  produce  rotation  outward,  and 


20G  MECHANICS. 

the  weight  of  the  wall  acts  to  prevent  it.     In  order  that  these  forces 
may  be  in  equilibrium,  their  moments  must  be  equal ;  or, 

QOUit^  =  Ik^  X  10.4166. 
Whence, 

t  =  h  v/.1802  =  .36  X  h. 

Next,  to  prevent  sliding.  The  force  of  friction  due  to  the  weight 
of  the  wall,  is, 

leO^A^X  .Q  =  dmi; 
and  that  the  wall  may  not  slide,  this  must  be  equal  to  the  pressure 
exerted  horizontally  against  the  wall.     Hence, 

96mt  =  dl.25lli:\ 
Whence, 

t  =  .325/i. 

If  the  wall  is  thick  enough  to  prevent  rotation,  it  is  secure  against 
sliding. 

6.  What  must  be  the  thickness  of  a  rectangular  dam  15  feet  high, 
the  weight  of  the  material  being  140  lbs.  to  the  cubic  foot,  when  the 
water  rises  to  the  top,  that  the  structure  may  be  just  on  the  point  of 
overturning  ?  Am.  5.7  ft. 

7.  The  staves  of  a  cylindrical  cistern  filled  with  water,  are  held 
together  by  a  single  hoop.     Where  should  the  hoop  be  situated  ? 

Arts.  At  a  distance  from  the  bottom  equal  to  one-third  of  the 
height  of  the  cistern. 

8.  Required  the  pressure  of  the  sea  on  the  cork  of  an  empty  bottle, 
when  sunk  to  the  depth  of  600  feet,  the  diameter  of  the  cork  being 
I  of  an  inch,  and  a  cubic  foot  of  sea-water  weighing  64  lbs.  ? 

Am.  134  lbs. 

Buoyant  Sffort  of  Fluids. 

148.  Let  ^  be  a  solid,  suspended  in  a  fluid.  Conceive  it 
divided  into  vertical  prisms,  whose  horizontal  sections  are 
infinitely  small.  Each  prism  is  pressed  down  by 
a  force  equal  to  the  weight  of  a  column  of  fluid, 
whose  base,  (Art.  146),  is  the  horizontal  section  of 
the  filament,  and  whose  altitude  is  the  distance 


Ag) 


of  its  upper  surface  from  the  surface  of  the  fluid;     Fig.  124. 
it  is  pressed  up  by  a  force  equal  to  the  weight  of  a  column 
of  fluid  having  the  same  base,  and  an  altitude  equal  to  the 


MECHANICS    OF    LIQUIDS.  207 

distance  of  the  lower  base  of  the  filament  from  the  surface 
of  the  fluid.  The  resultant  of  these  pressures  is  exerted 
vertically  ujjward,  and  is  equal  to  the  weight  of  a  column 
of  the  fluid,  equal  in  bulk  to  that  of  the  filament,  and 
having  its  point  of  application  at  the  centre  of  gravity  of 
the  filament;  the  lateral  pressures  destroy  each  other's 
effects ;  hence,  the  resultant  pressure  on  the  body,  is  a  ver- 
tical force  exerted  upward,  whose  intensity  is  equal  to  the 
weight  of  the  displaced  fluid,  and  whose  point  of  applica- 
tion is  the  centre  of  gravity  of  the  displaced  fluid.  This 
upward  pressure  is  the  buoyant  effort  of  the  fluid,  and  its 
point  of  application  is  the  centre  of  buoyancy.  The  direction 
of  the  buoyant  effort,  in  any  position  of  the  body,  is  a  line 
of  support.  That  line  of  support  which  passes  through  the 
centre  of  gravity,  of  the  body  is  the  line  of  rest. 

Floating  Bodies. 

149.  A  body  immersed  in  a  fluid,  is  urged  downward  by 
its  weight  applied  at  its  centre  of  gravity,  and  upward,  by 
the  buoyant  effort  of  the  fluid  applied  at  the  centre  of 
buoyancy. 

The  body  can  only  be  in  equilibrium  when  the  line 
through  the  centre  of  gravity  of  the  bod}^,  and  the  centre 
of  buoyancy,  is  vertical ;  in  other  words,  when  the  line  of 
rest  is  vertical.  When  the  weight  of  the  body  exceeds  the 
buoyant  effort,  the  body  sinks  to  the  bottom ;  when  they 
are  equal,  it  remains  in  equilibrium,  wherever  placed. 
When  the  buoyant  effort  is  greater  than  the  weight,  it  rises 
to  the  surface,  and,  after  a  few  oscillations,  comes  to  rest, 
in  such  a  position,  that  the  weight  of  the  displaced  fluid  is 
equal  to  that  of  the  body,  when  it  is  said  to  float.  The 
upper  surface  of  the  fluid  is  then  called  the  plane  of  flota- 


208 


MECHANICS. 


W 


Fig.  125. 


Fig.  126. 


tion,  and  its  intersection  with  the  surface  of  the  body,  ^he 
line  of  flotation. 

If  a  floating  body  be  slightly  disturbed  from  its  position 
of  equilibrium,  the  centres  of 
gravity  and  buoyancy  are  no  longer 
in  the  same  vertical.  Let  DE  re- 
present the  plane  of  flotation,  G 
the  centre  of  gravity  of  the  body, 
(Fig.  126),  GH  its  line  of  rest, 
and  C  the  centre  of  buoyancy. 

If  the  line  of  support,  CB,  in- 
tersect the  line  of  rest  in  M,  above 
G^  as  in  Fig.  126,  the  buoyant 
effort  and  the  weight  conspire  to 
restore  the  body  to  equilibrium ; 
in  this  case,  the  equilibrium  is  ddble. 

If  M  is  below  G,  as  in  Fig.  127,  the  buoyant  effort  and 
the  weight  conspire  to  overturn  the  body;  in  this  case  the 
body,  before  being  disturbed,  must 
have  been  in  unstable  equilibrium. 

If  the  centres  of  buoyancy  and  grav- 
ity are  always  on  the  same  vertical,  M 
coincides  with  G  (Fig.  128),  and  the 
body  is  in  indifferent  equilibrium.  The 
limiting  position  of  J/,  obtained  by  de- 
flecting the  body  through  an  infinitely 
small  angle,  is  the  metacentre  of  the 
body.     Hence, 

If  the  metacentre  is  above  the  centre 
of  gravity,  the  body  is  in  stable  eqni 
lihrium  ;  if  below  the  centre  of  gravity ,  ^^^"  ^'*' 

the  body  is  in  unstable  cquiUbrimn;  if  tlie  poirits  coincide, 
the  body  is  in  indifferent  equilibrium. 


MECHANICS    OF   LIQUIDS.  209 

The  stability  of  the  floating  body  is  greater,  as  the  meta- 
centre  is  liigher  above  the  centre  of  gravity.  This  con^ 
dition  is  fulfilled  in  loading  ships,  by  stowing  the  heavier 
objects  near  the  bottom  of  the  vessel. 

Specific  Gravity. 

150.  The  specific  gravity  of  a  body  is  its  relative  weight, 
that  is,  it  is  the  number  of  times  the  body  is  heavier  than 
an  equivalent  volume  of  some  other  body,  taken  as  a 
standard. 

The  specific  gravity  of  a  body  is  obtained  by  dividing 
the  weight  of  any  volume  of  the  body,  by  that  of  an  equiv- 
alent volume  of  the  standard. 

For  solids  and  liquids,  distilled  water  is  taken  as  a 
standard.  Because  this  liquid  is  of  different  densities  at 
different  temperatures,  it  becomes  necessary  to  assume  a 
standard  temperature  for  it :  for  a  like  reason,  a  standard 
temperature  must  be  taken  for  the  body  whose  specific 
gravity  is  to  be  found.  Different  standards  of  temperature 
have  been  assumed  by  diff'erent  writers;  we  shall  adopt 
those  assumed  by  Jamin",  who  takes  for  the  standard  tem- 
perature of  water,  4°  C,  or  about  39"  F.,  and  for  the  stand- 
ard temperature  of  the  body,  0°  C,  or  32°  F.  The  former 
is  the  temperature  at  which  water  has  a  maximum  density, 
and  the  latter  is  that  of  melting  ice. 

In  finding  the  specific  gravity  of  a  body,  we  first  deter- 
mine it  with  respect  to  water  at  any  temperature ;  this  we 
may  call  the  observed  specific  gravity.  We  then  correct  the 
result  for  the  temperature  of  the  w^ater,  by  means  of  a 
table  of  densities  of  water  at  different  temperatures,  that 
at  39°  being  1 ;  this  result  we  call  the  apparent  specific 
gravity.  Finally,  we  correct  this  for  the  temperature  of 
the  body,  and  thus  find  the  true  specific  gravity. 


^10  MECHANICS. 

1st.  Let  d  be  the  density  of  water  at  the  temperature,  /, 
its  density  at  39°  being  1 ;  let  s  be  the  observed  specific 
gravity  of  a  body  referred  to  water  at  the  temperature,  /, 
and  let  s'  be  its  specific  gravity  referred  to  water  at  39°. 

Because  the  specific  gravity  of  a  body  varies  inversely  as 
the  density  of  the  water  to  whicli  it  is  referred,  we  have, 

s  '.  s'  '.:  1  :  d;     .'.     s'  =  ds. 

That  is,  to  find  the  a2:i2^arent  sjjecific  gravity  of  a  body, 
multiply  its  observed  s^jecific  gravity,  at  the  temperature,  t, 
by  the  corresponding  tabular  density  of  water. 

2dly.  Suppose  the  body  to  have  the  same  temperature,  t, 
as  the  water  to  which  it  is  referred.  Denote  the  volume 
of  the  body  at  the  temperature,  t,  by  v',  and  at  32°,  by  v; 
denote  the  corresponding  specific  gravities  by  s'  and  5. 

Because  the  specific  gravity  varies  inversely  as  its  vol- 
ume, we  have, 

,v' 

s  :  s    :  :  V    :  v;      .*.    s  =s  -. 

V 

That  is,  tofi7id  the  true  specific  gravity  of  a  body,  multiply 
its  ap2mrent  specific  gravity  by  the  quotient  of  its  volume 
at  the  temperature,  t,  by  its  volume  at  32°. 

This  quotient  may  be  found  from  the  body's  known  rate 
of  expansion. 

It  is  only  in  nice  determinations  that  it  is  necessary  to 
take  account  of  the  latter  correction. 

Gases  are  usually  referred  to  air  as  a  standard  ;  but  as  air 
is  easily  referred  to  water,  we  may  take  distilled  water  at 
39°  F.  as  a  standard  for  all  bodies. 

Sometimes  it  is  convenient  to  find  the  specific  gravity 
of  a  body  with  respect  to  some  other  body  whose  specific 
gravity  is  already  known.  In  this  case  the  required  spe- 
cific gravity  is  equal  to  the  product  of  that  which  is  found, 


MECHANICS   OF    LIQUIDS.  211 

by  that  which  is  already  known.  Thus,  if  A  is  m  times  as 
heavy  as  B,  and  if  B  is  n  times  as  heavy  as  C,  then  will  A 
be  71171  times  as  heavy  as  C. 

Methods  of  finding  Specific  Gravity. 

151.  There  are  two  principal  methods  of  finding  the  spe- 
cific gravity  of  a  body ;  first,  by  means  of  the  balance,  and 
secondly,  by  means  of  the  hydrometer.  The  former  alone 
can  be  used  for  nice  determinations,  such  as  are  needed  in 
the  operations  of  analytical  chemistry ;  the  latter  is  of 
easier  application,  and  is  sufficiently  accurate  for  most  prac- 
tical purposes. 

Hydrostatic  Balance. 

152.  This  balance  is  similar  to  that  described  in  Arti- 
cle 68 ;  the  scale-pans,  however,  are  provided  with  hooks 
for  suspending  bodies,  as  shown  in 

the  figure. 

In  balances  of  modern  construc- 
tion the  vessel  containing  water  is 
placed  on  a  movable  bench  or  shelf, 
that  strides  one  of  the  scale-pans,   ~ 

Fig.  129. 

without  interfering  with  its  move- 
ments, and  the  body  is  then  suspended  from  the  beam  by  a 
thread  or  wire.     In  both  cases  a  body  attached  to  the  string 
maybe  weighed  either  in  the  air  or  in  the  water,  at  pleasure. 

Specific  Gravity  of  an  Insoluble  Body. 

153.  Fasten  the  suspending  wire  to  one  scale-pan,  or  to 
one  extremity  of  the  beam,  as  the  case  may  be,  and  coun- 
terpoise it  by  weights  in  the  opposite  pan.  Then  attach 
the  body  to  the  wire  and  counterpoise  it  by  weights  in  the 
other  pan:  these  give  the  weight  of  the  body  in  air:  next 
immerse  the  body  in  water,  so  as  not  to  touch  the  contain- 


A 


1 


212  MECHANICS. 

I'ng  vessel ;  the  buoyant  effort  of  the  water  will  th rnst  the 
body  up  with  a  force  equal  to  the  weight  of  the  displaced 
water:  restore  the  equilibrium  by  weights  placed  in  the 
first  pan  ;  these  will  give  the  tveight  of  the  disjylaced  water : 
divide  the  weight  of  the  body  in  air  by  the  weight  of  the 
.displaced  water,  and  the  quotient  will  be  the  observed  spe- 
cific gravity. 

Thus,  if  a  piece  of  copper  weigh  2047  grains  in  air,  and 
lose  230  grains  when  weighed  in  water,  its  specific  gravity 
is  VA^  or  8.9. 

If  the  body  will  not  sink  in  water,  determine  its  weight 
in  air,  as  before;  then  attach  to  it  a  body  so  heavy  that  the 
combination  will  sink ;  find  the  weight  of  the  water  dis- 
placed by  the  combination,  and  also  the  weight  displaced 
by  the  heavy  body,  take  their  difference,  and  the  result  will 
be  the  weight  of  the  water  displaced  by  the  body  in  ques- 
tion ;  then  proceed  as  before. 

Thus,  a  body  weighs  600  grains  in  air ;  when  attached  to 
a  piece  of  copper,  the  combination  weighs  2647  grains  in 
air,  and  suff"ers  a  loss  of  834  grains  in  water,  the  copper 
alone  losing  230  grains.  The  buoyant  eff'ort  of  the  fluid 
exerted  on  the  body  is  therefore  604  grains,  and  the  specific 
gravity  of  the  body  is  JQf,  or  0.993. 

Specific  Gravity  of  a  Soluble  Body. 

154.  Find  its  specific  gravity  with  respect  to  some  liquid 
in  which  it  is  not  soluble ;  find  also  the  specific  gravity  of 
this  liquid  with  respect  to  water;  take  the  product  of 
these,  and  it  will  be  the  specific  gravity  sought,  (Art.  150). 

Thus,  if  the  specific  gravity  of  a  body  with  respect  to 
oil  be  3.7,  and  the  specific  gravity  of  the  oil  with  respect 
to  water  be  0.9,  the  specific  gravity  of  the  body  is  3.7  X  0.9, 
or  3.33. 


MECHANICS    OF    LIQUIDS.  213 

It  is  often  convenient  to  use  a  saturated  solution  of  the 
substance  in  question  as  the  auxiliary  liquid. 

Specific  G-ravity  of  Liquids. 

155.  16?^^  Method. — The  most  convenient  method  is  by 
the  specific  gravity  bottle.  This  is  a  bottle  constructed  to 
hold  exactly  1000  grains  of  distilled  water.  Accompany- 
ing it  is  a  brass  weight,  just  equal  to  the  empty  bottle. 
To  use  it,  let  it  be  filled  with  the  liquid  in  question,  and 
placed  ill  one  scale-pan ;  in  the  other  pan  place  the  brass 
counterpoise,  and  weights  enough  to  balance  the  liquid; 
divide  the  number  of  grains  in  the  weight  of  the  liquid 
by  1000,  and  the  quotient  will  be  the  specific  gravity. 

Thus,  if  the  bottle  filled  with  a  liquid  weighs  945  grains, 
beside  the  counterpoise,  its  specific  gravity  is  0.945. 

2(1  Method. — Take  u  body,  that  Avill  sink  both  in  the 
liquid  and  in  water,  and  which  is  not  acted  upon  by  either; 
determine  its  loss  of  weight,  first  in  the  liquid,  then  in 
water;  divide  the  former  by  the  latter,  and  the  quotient 
will  be  the  specific  gravity  sought.  The  reason  is 
evident. 

Thus,  if  a  glass  ball  lose  30  grains  when  weighed  in 
water,  and  24  in  alcohol,  the  specific  gravity  of  the  alcohol 
is  II,  or  0.8. 

Zd  Method. — Let  AB  and  CD  be  graduated  glass  tubes, 
half  an  inch  in  diameter,  open  at  both  ends.  Let  their 
upper  ends  communicate  with  the  receiver 
of  an  air-pump,  and  their  lower  ends  dip 
into  two  vessels,  one  containing  distilled 
water,  and  the  other  the  liquid  whose  specific 

gravity  is  to  be  determined.      Let  the  air  be        "Jl IJE 

partially  exhausted  from  the  receiver  by  an    — ^   '  '^' 
air-pump ;  the  liquid  will  rise  in  the  tubes         ^^^'  ^^' 


214 


MECHANICS. 


to  heights  inversely  as  the  specific  gravities  of  the  liquids. 
If  we  divide  the  height  of  the  column  of  water  by  that  of 
the  other  liquid,  the  quotient  will  be  the  specific  gravity 
sought.  By  producing  different  degrees  of  rarefaction, 
the  columns  will  rise  to  different  heights,  but  their  ratios 
ought  to  be  the  same.  "We  are  thus  enabled  to  make  a 
series  of  observations,  each  corresponding  to  a  different 
degree  of  rarefaction,  from  which  a  more  accurate  result 
can  be  had,  than  from  a  single  observation. 


Specific  Gravity  of  Air. 

156.  Take  a  globe,  fitted  with  a  stop-cock,  and,  by  means 
of  an  air-pump,  or  condensing  syringe,  force  in  as  much 
air  as  is  convenient,  close  the  stop-cock,  and  weigh  the 
globe  thus  filled.  Provide  a  glass  tube,  graduated  to  cubic 
inches  and  decimals  of  a  cubic  inch, 
and,  having  filled  it  with  mercury, 
invert  it  over  a  mercury  bath.  Open 
the  stop-cock,  and  allow  the  compressed 
air  to  escape  into  the  tube,  taking  care 
to  place  the  tube  in  such  a  position  that 
the  mercury  without  the  tube  is  at  the 
same  level  as  within.  The  reading  on  the  tube  gives  the 
volume  of  escaped  air.  Weigh  the  globe  again,  and  sub- 
tract the  weight  thus  found  from  the  first  weight;  this 
difference  is  the  weight  of  the  escaped  air.  Having 
reduced  the  measured  volume  of  air  to  what  it  would  have 
occupied  at  a  standard  temperature  and  pressure,  by  rules 
yet  to  be  deduced,  compute  the  weight  of  an  equivalent 
volume  of  water ;  divide  the  weight  of  the  corrected  volume 
of  air  by  that  of  an  equivalent  volume  of  distilled  water, 
and  the  quotient  will  be  the  specific  gravity  sought. 


Fig.  131. 


MECHANICS   OF   LIQUIDS. 


216 


Hydrometers. 

157.  A  hydrometer  is  a  floating  body,  used  in  finding 
specific  gravities.  Its  construction  depends  on  the  prin- 
ciple of  flotation.  Hydrometers  are  of  two  kinds.  1°.  Those 
in  which  the  submerged  volume  is  constant.  2°.  Those  in 
which  the  weight  of  the  instrument  is  constant. 

Nicholson's   Hydrometer. 

158.  This  instrument  consists  of  a  hollow  cylinder,  A, 
at  the  lower  extremity  of  which  is  a  basket,  B,  and  at  the 
upper  extremity  a  wire,  bearing  a  scale-pan,  C. 
At  the  bottom  of  the  basket  is  a  ball,  F,  con- 
taining mercury,  to  cause  the  instrument  to 
float  in  an  upright  position.  By  means  of  this 
ballast,  the  instrument  is  adjusted  so  that  a 
given  weight,  say  500  grains,  placed  in  the  pan, 
C,  will  sink  it  in  distilled  water  to  a  notch,  D, 
filed  in  the  neck. 

This   instrument   is   in    reality  a  weighing- 
machine,  and  as  such  can  be  used  for  determin-       F^g- 132. 
ing  the  approximate  weights  of  bodies  Avi thin  certain  limits; 
in  the  instrument  described,  no  body  can  be  weighed  whose 
weight  exceeds  500  grains. 

To  find  the  specific  gravity  of  a  solid,  place  it  in  the  pan, 
C,  and  add  weights  till  the  instrument  sinks,  in  distilled 
water,  to  the  notch,  D.  The  added  weights,  subtracted 
from  500  grains,  give  the  weight  of  the  body  in  air.  Place 
the  body  in  the  basket,  B,  which  generally  has  a  reticu- 
lated cover,  to  prevent  the  body  from  floating  away,  and 
add  other  weights  to  the  pan,  until  the  instrument  again 
sinks  to  the  notch,  D.  The  weights  last  added  give  the 
weight  of  water  displaced  by  the  body.     Divide  the  first  of 


216  MECHANICS. 

these  by  the  second,  and  the  quotient  will  be  the  specific 
gravity  required. 

To  find  the  specific  gravity  of  a  liquid.  Having  weighed 
the  instrument,  place  it  in  the  liquid,  and  add  weights  to 
the  scale-pan,  till  it  sinks  to  D.  The  weight  of  the  instru- 
ment, plus  the  weights  added,  will  be  the  weight  of  the 
liquid  displaced  by  the  instrument.  The  weight  of  the  in- 
strument added  to  500  grains  gives  the  weight  of  an  equal 
volume  of  distilled  water.  The  quotient  of  the  first  by  the 
second  is  the  specific  gravity  required. 

A  modification  of  this  instrument,  in  which  the  basket, 
B,  is  omitted,  is  sometimes  used  for  determining  specific 
gravities  of  liquids  only.  This  kind  of  hydrometer,  known 
as  Fahrenheit's  hydrometer,  is  generally  made  of  glass,  that 
it  may  not  be  acted  on  chemically  by  the  liquids  into  which 
it  is  plunged. 

Scale   Areometer. 

159.  The  scale  areometer  is  a  hydrometer  whose  weight 
is  constant;  the  specific  gravity  of  a  liquid  is  made  known 
by  the  depth  to  which  it  sinks  in  it.  The  instru- 
ment  consists  of  a  glass  cylinder,  J,  with  a  stem,  C, 
of  uniform  diameter.  At  the  bottom  of  the  cylin- 
der is  a  bulb,  B,  containing  mercury,  to  make  the 
instrument  float  upright.  By  introducing  a  suit- 
able quantity  of  mercury,  the  Instrument  may  be 
adjusted  so  as  to  float  at  any  desired  point  of  the 
stem. 

When  it  is  designed  to  determine  the  specific  B' 
gravity  of  liquids,  both  lighter  and  heavier  than  Fig.  133. 
distilled  water,  it  is  called  a  universal  hydrometer ,  and  is 
so  ballasted  as  to  float  in  distilled  water  at  the  middle  of 
the  stem.     This  point  is  marked  on  the  stem  with  a  file, 


H 

t  E 


„y 


MECHANICS    OF    LIQUIDS.  217 

and  is  numbered  1  on  the  scale.  A  liquid  is  then  formed, 
by  dissolving  salt  in  water,  whose  specific  gravity  is  1.1, 
and  the  instrument  is  allowed  to  float  freely  in  it;  the 
pointj  E,  to  which  it  sinks,  is  marked  on  the  stem,  and  the 
intermediate  part  of  the  scale,  HE^  is  divided  into  10  equal 
parts.  In  like  manner  a  mixture  of  alcohol  and  water  is 
formed,  whose  specific  gravity  is  0.9,  the  corresponding 
position  of  the  plane  of  flotation  is  marked  on  the  stem, 
and  the  space  between  it  and  the  division  1  is  divided  into 
10  equal  parts.  The  graduation  is  continued,  both  up  and 
down,  through  the  whole  length  of  the  stem.  The  gradua- 
tion is  marked  on  a  piece  of  paper  within  the  stem.  To 
use  this  hydrometer,  we  put  it  into  the  liquid  and  allow  it 
to  come  to  rest ;  the  division  of  the  scale  that  correspords 
to  the  surface  of  flotation  shows  the  specific  gravity  of  the 
liquid.  The  hypothesis  on  which  this  instrument  is  gradu- 
ated, is,  that  the  increments  of  specific  gravity  are  pro- 
portional to  the  increments  of  the  submerged  portion  of 
the  stem.  This  hypothesis  is  only  approximately  true,  but 
it  approaches  more  nearly  to  the  truth  as  the  diameter  of 
the  stem  diminishes. 

When  it  is  only  desired  to  use  the  instrument  for  liquids 
heavier  than  water,  the  instrument  is  ballasted  so  that  the 
division  1  shall  be  near  the  top  of  the  stem.  If  it  is  to  be 
used  for  liquids  lighter  than  water,  it  is  ballasted  so  that 
the  division  1  shall  be  near  the  bottom  of  the  stem.  In 
this  case  we  determine  the  point  0.9  by  using  a  mixture  of 
alcohol  and  water,  the  j^rmciple  of  graduation  being  the 
same  as  in  the  first  instance. 

Volumeter. 

160.  The  volumeter  is  a  modification  of  the  scale  areom- 
eter, differing  from  it  only  in  graduation.     The  gradua- 

10 


E 


E 


218  MECHANICS. 

tion  is  effected  as  follows:  The  instrument  is  placed  in 
distilled  water,  and  allowed  to  come  to  rest,  and  the  point 
of  the  stem  where  the  surface  cuts  it,  is  marked 
with  a  file.  The  submerged  volume  is  then  accu- 
rately determined,  and  the  stem  is  graduated  in 
such  manner  that  each  division  indicates  a  volume 
equal  to  a  hundredth  part  of  the  volume  originally 
submerged.  The  divisions  are  then  numbered  from 
the  first  mark  in  both  directions,  as  indicated  in 
the  figure.  To  use  the  instrument,  place  it  in  the 
liquid,  and  note  the  division  to  which  it  sinks:     ^^ 

^         '  '  Fig.  134 

divide  100  by  the  number  indicated,  and  the  quo- 
tient will  be  the  specific  gravity  sought.  The  principle 
employed  is,  that  the  specific  gravities  of  liquids  are  in- 
versely as  the  volumes  of  equal  weights.  Suppose  that 
the  instrument  indicates  x  parts;  then  the  weight  of  the 
instrument  displaces  x  parts  of  the  liquid,  whilst  it  dis- 
places 100  parts  of  water.  Denoting  the  specific  gravity 
of  the  liquid  by  8,  and  that  of  water  by  1,  we  have, 

S  :  1  '.:  100  '.  X,     .-.  S  ^  — . 

X 

A  table  may  be  computed  to  save  performing  the  divi- 
Bion. 

Densimeter. 

161.  The  densimeter  admits  of  use  when  only  a  small 
portion  of  the  liquid  can  be  had.  Its  construction  differs 
from  that  of  the  volumeter,  in  having  a  small  cup  at  the 
upper  extremity  of  the  stem,  to  receive  the  fluid  whose 
specific  gravity  is  to  be  determined. 

The  instrument  is  so  ballasted  that  when  the  cup  is 
empty,  the  densimeter  sinks  in  distilled  water  to  a  point, 


MECHANICS   OF    LIQUIDS.  219 

B,  near  the  bottom  of  the  stem.     This  point  is  the  0  of 
the  instrument.    The  cup  is  then  filled  with  distilled  water, 
and  the  point,  G,  to  which  it  sinks,  is  marked;  the 
space,  BC,  is  divided  into  any  number  of  equal       ^ 
parts,  say  10,  and  the  graduation  is  continued  to 
the  top  of  the  tube.  ^ 

To  use  the  instrument,  place  it  in  distilled  water, 
and  fill  the  cup  with  the  liquid  in  question,  and  ^ 
note  the  division  to  which  it  sinks.  Divide  the 
number  of  this  division  by  10,  and  the  quotient 
will  be  the  specific  gravity  required.  The  principle 
of  the  densimeter  is,  that  the  specific  gravity  of  a 
body  of  a  constant  volume  is  proportional  to  the  ^^s-  ^^s- 
volume  of  water  it  causes  the  instrument  to  displace.  • 


Centesimal   Alcoometer  of  Gay  Lussac. 

162.  This  instrument  is  similar  in  construction  to  the 
scale  areometer;  the  graduation,  however,  is  made  on  a 
different  principle.  Its  object  is,  to  determine  the  percent- 
ao^e  of  alcohol  in  a  mixture  of  alcohol  and  water.  The 
graduation  is  made  as  follows:  the  instrument  is  first 
placed  in  absolute  alcohol,  and  ballasted  so  that  it  will 
sink  nearly  to  the  top  of  the  stem.  This  point  is  marked 
100.  Next,  a  mixture  of  95  parts  of  alcohol  and  5  of 
water,  is  made,  and  the  point  to  which  the  instrument 
sinks,  is  marked  95.  The  intermediate  space  is  divided 
into  5  equal  parts.  Next,  a  mixture  of  90  parts  of  alcohol 
and  10  of  water  is  made ;  the  point  to  which  the  instru- 
ment sinks,  is  marked  90,  and  the  space  between  this  and 
95,  is  divided  into  5  equal  parts.  In  this  manner,  the 
entire  stem  is  graduated  by  successive  operations.  The 
spaces  on  the  scale  are  not  equal  at  different  points,  but, 


220  MECHANICS. 

for  a  space  of  five  parts,  they  may  be  so  regarded,  without 
sensible  error. 

To  use  the  instrument,  place  it  in  the  mixture  of  alcohol 
and  water,  and  read  the  division  to  which  it  sinks;  this 
will  indicate  the  percentage  of  alcohol  in  the  mixture. 

In  all  the  instruments,  the  temperature  has  to  be  taken 
into  account;  this  is  effected  by  tables  that  accompany  the 
different  instruments. 

On  the  principle  of  the  alcoometer,  a  great  variety  of 
areometers  are  constructed,  for  determining  the  strength 
of  wines,  syrups,  and  other  liquids  employed  in  the  arts. 

In  some  nicely  constructed  hydrometers,  the  mercury 
used  as  ballast  serves  also  to  fill  the  bulb  of  a  delicate 
thermometer,  whose  stem  rises  into  the  cylinder  of  the 
instrument,  and  thus  enables  us  to  note  the  temperature 
of  the  fluid  in  which  it  is  immersed. 

Examples. 

1.  A  cubic  foot  of  water  weighs  1000  ounces.  Required  the  weight 
of  a  cubical  block  of  stone,  whose  edge  is  4  feet,  its  specific  gravity 
being  2.5.  Anf<.  10000  lbs. 

2.  Required  the  number  of  cubic  feet  in  a  body  whose  weight  is 
1000  lbs.,  its  specific  gravity  being  1.25.  Ans.  12.8. 

3.  Two  lumps  of  metal  weigh  3  lbs.,  and  1  lb.,  and  their  specific 
gravities  are  5  and  9.  What  will  be  the  specific  gravity  of  an  alloy 
formed  by  melting  them  together,  supposing  no  ccmtraction  of 
volume  to  take  place?  Am.  5.625. 

4.  A  body  weighing  20  grains  has  a  specific  gravity  of  2.5.  Re- 
quired Its  loss  of  weight  in  water.  Ans.  8  grains. 

5.  A  body  weighs  25  grains  in  water,  and  40  grains  in  a  liquid 
whose  specific  gravity  is  .7.  What  is  the  weight  of  the  body  in 
vacuum?  Am.  75  grains. 

6.  A  Nicholson's  hydrometer  weighs  250  grains,  and  it  requires 
an  additional  weight  of  336  grains  to  sink  it  to  the  notch  in  the  stem, 
in  a  mixture  of  afcohol  and  water.  What  is  the  specific  gravity  of 
the  mixture?  Am.  .781. 


MECHANICS    OF    LlC^L'lDt?. 


2^1 


7.  A  block  of  wood  sinks  in  distilled  water  till  I  of  its  volume  is 
submerged.     What  is  its  specific  gravity?  Ans.  .875. 

8.  The  weight  of  a  piece  of  cork  in  air,  is  |  oz. ;  the  weight  of  a 
piece  of  lead  in  water,  is  6^  oz.  •,  the  weight  of  the  cork  and  lead 
together  in  water,  is  4-ili,  oz.  What  is  the  specific  gravity  of  the 
cork?  Ans.  0.24. 

9.  A  solid,  whose  weight  is  250  grains,  weighs  in  water,  147  grains, 
and,  in  another  fluid,  120  grains.  What  is  the  specific  gravity  of  the 
latter  fluid?  Ans.  1.262. 

10.  A  solid  weighs  60  grains  in  air,  40  in  water,  and  30  in  an  acid. 
What  is  the  specific  gravity  of  the  acid  ?  Ans.  1.5. 

The   following   table   is   compiled   from   the  Ordnance 
Manual. 

TABLE    OP    SPECIFIC    GRAVITIES   OF    SOLIDS    AND    LIQUIDS. 


SOI.IDS. 

SPKC.  GllAV. 

SOLIDS. 

SPEC.  GRAV. 

3.180 
2.686 
2.130 
1.800 
2.612 
2.520 
0.945 
0.912 
0.596 
0.715 
1.333 
0.854 
1.170 
0  660 
1.217 
1.841 
0.792 
0.715 
1.026 
0.915 
0.870 

Antimony,  cast 

Brass,  cast 

Copper,  cast 

Gold,  hammered. . . 

Iron,  bar 

Iron,  cast  

Lead,  cast 

Mercury  at  32°  F. . 
at  60°.... 

Platina,  rolled 

cast 

Silver,  hammered. . 

Tin,  cast 

Zinc,  cast 

Bricks 

Chalk 

Coal,  bituminous  . . 

Diamond 

Earth,  common  ... 

Gypsum 

Ivory 

6.712 

8.396 

8.788 

19.361 

7.788 

7.207 

11.352 

13.598 

13.580 

22.069 

20.337 

10  511 

7.291 

6.861 

1.900 

2.784 

1.270 

3.521 

1.500 

2.168 

1.822 

Limestone 

Marble,  comm(m . . . 

Salt,  common 

Sand 

Slate 

Stone,  comn.on .... 

Tallow 

Boxwood 

Cedar 

Cherry 

Lignum  vitae 

Mahoganv 

Oak,  heart 

Pine,  yellow 

Nitric  acid 

Sulphuric  acid 

Alcohol,  absolute. . . 
Ether,  sulphuric. .. 
Sea  water 

Olive  oil 

Oil  of  Turpentine. . 

Thermometer. 
163.  A  THERMOMETER,  is  till  instrument  for  measuring 
the  temperatures  of  bodies.    All  bodies  expand  when  heated, 


222  MECHANICS. 

and  contract  when  cooled,  and,  other  things  being  equal, 
always  occupy  the  same  volumes  at  the  same  temperatures. 
Different  bodies  expand  and  contract  in  different  ratios 
for  equal  increments  of  temperature.  As  a  general  rule, 
liquids  expand  more  rapidly  than  solids,  and  gases  more 
rapidly  than  liquids.  The  construction  of  the  thermom- 
eter depends  on  this  principle  of  unequal  expansibility  of 
bodies.  A  great  variety  of  forms  have  been  used,  only  one 
of  which  will  be  described. 

The  mercurial  thermometer  consists  of  a  bulb.  A,  at  the 
upper  extremity  of  which  is  a  tube  of  uniform  bore, 
hermetically  sealed  at  its  upper  end.  The  bulb  ^_^ 
and  tube  are  nearly  filled  with  mercury,  and  to 
the  whole  is  attached  a  frame,  on  which  is  a  scale 
for  temperature. 

A  thermometer  may  be  constructed  as  follows: 
A  tube  of  uniform  bore  is  selected,  and  on  one 
extremity  a  bulb  is  blown,  which  may  be  cylin- 
drical, or  spherical ;  the  former  shape  is,  on  many 
accounts,  the  preferable  one.  At  the  other  ex- 
tremity, a  conical-shaped  funnel  is  blown,  open  at 
top.  The  funnel  is  filled  with  mercury,  which 
should  be  of  the  purest  quality,  and  the  whole  p/'^ 
being  held  vertical,  the  heat  of  a  spirit-lamp  is 
applied  to  the  bulb,  which  expanding  the  air  contained  in 
it,  forces  a  portion  in  bubbles  up  througli  the  mercury  in 
the  funnel.  The  instrument  is  next  alloAved  to  cool,  when 
a  portion  of  mercury  is  forced  down  the  tube  into  the 
bulb.  By  a  repetition  of  this  process,  the  entire  bulb  may 
be  filled  with  mercury,  as  well  as  the  tube  itself  Heat  is 
then  applied  to  the  bulb,  until  the  mercury  is  made  to 
boil;  and,  on  being  cooled  down  to  a  little  above  the 
highest   temperature  that  it   is   desired  to   measure,   the 


■  0 

9 


MECHAJS^ICS   OF   LIQUIDS.  223 

top  of  the  tube  is  melted  off  by  a  jet  of  flame,  urged  by  a 
blow-pipe,  and  the  whole  hermetically  sealed.  The  instru- 
ment, thus  prepared,  is  attached  to  a  frame,  and  graduated 
as  follows : 

The  instrument  is  plunged  into  a  bath  of  melting  ice, 
and,  after  remaining  a  sufficient  time  for  the  instrument  to 
take  the  temperature  of  the  ice,  the  height  of  the  mercury 
in  the  tube  is  marked  on  the  scale.  This  gives  the  freezing 
point.  The  instrument  is  next  plunged  into  a  bath  of 
boiling  water,  and  allowed  to  remain  long  enough  to 
acquire  the  temperature  of  the  water  and  steam.  The 
height  of  the  mercury  is  then  marked  on  the  scale.  This 
gives  the  boiling  point.  The  freezing  and  boiling  points 
having  been  determined,  the  intermediate  space  is  divided 
into  a  certain  number  of  equal  parts,  according  to  the 
scale  adopted,  and  the  graduation  is  continued,  both  up 
and  down,  to  any  desired  extent. 

Three  principal  scales  are  used.  Fahrenheit's  scale, 
in  which  the  space  between  the  freezing  and  boiling  point 
is  divided  into  180  equal  parts,  called  degrees,  the  freezing 
point  being  marked  32°,  and  the  boiling  point  212°.  In 
this  scale,  the  0  point  is  3^  degrees  below  the  freezing 
point.  Tlic  Centigrade  scale,  in  which  the  space  between 
the  fixed  points  is  divided  into  100  equal  parts,  called 
degrees.  The  0  of  this  scale  is  at  the  freezing  point. 
Reaumur's  scale,  in  which  the  same  space  is  divided  into 
80  equal  parts,  called  degrees.  The  0  of  this  scale  also  is 
at  the  freezing  point. 

If  we  denote  the  number  of  degrees  on  the  Fahrenheit, 
Centigrade,  and  Reaumur  scales,  by  F,  C,  and  R  respect- 
ively, the  following  formula  will  enable  us  to  pass  from 
any  one  of  these  scales  to  any  other : 

i(i^°  -  32°)  =  iC'°  =  iR° 


224:  MECHANICS 

The  scale  most  in  use  in  this  country  is  Fahrenheit's. 
The  other  two  are  used  in  Europe,  particularly  the  Centi- 
grade scale. 

Velocity  of  a  Liquid  through  a  small  Orifice. 

164.  Let  ABD  be  a  vessel,  having  a  small  orifice  at  its 
bottom,  and  filled  with  a  liquid. 

Denote  the  cross  section  of  the  orifice,  by  a,  and  its 
depth  below  the  upper  surface,  by  h.  Let  D  be  an  infi- 
nitely small  layer  of  the  liquid  at  the  orifice,  and 
denote  its  height,  by  W.  This  layer  is  (Art.  146) 
urged  downward  by  a  force  equal  to  the  weight  of 
a  column  of  the  liquid  whose  base  is  the  orifice, 
and  whose  height  is  h  ;  denoting  this  pressure,  by 
p,  and  the  weight  of  a  unit  of  volume   of  the 

liquid,  by  w,  we  have, 

p  =  wah. 

Were  the  element  pressed  downward  by  its  own  weight 

alone,  the  pressure  being  denoted  by  p',  we  should  have, 

p'  ~  wah'. 

Dividing  the  former  by  the  latter,  we  have, 

p  _  h 

that  is,  the  pressures  are  as  the  heights  h  and  h'. 

Let  us  suppose,  the  element  falls  through  the  height,  h\ 
first  under  the  action  of  the  force,  p,  and  then  under  the 
action  of  the  force,  p' .  Denoting  the  velocities  generated, 
by  V  and  v',  we  have,  (Art.  104), 


V  =  ^/2pli^  and,  v  —  v2p'h' ; 
whence,  by  reduction, 


V  :  v'  ::  \/p  :  Vp',   •' .    ^ 


MECHANICS    OF    LIQUIDS.  225 

But,  when  the  element  falls  under  the  action  of  v',or  its 
own  weight,  we  have, 

v'  =  V'igh'. 
Substituting  this  volume,  v',  and  replacing—,,  by  its  value, 

j-i,  we  have,  after  reduction, 

v=  V^gh. 

Hence,  a  liquid  issues  from  an  orifice  in  tlie  bottom  of  a 
vessel,  with  a  velocity  equal  to  that  acquired  hy  a  hody  in 
falling  through  a  height  equal  to  the  distance  of  the  orifice 
below  the  free  siirface. 

We  have  seen  that  the  pressure  due  to  the  weight  of  a 
fluid  on  any  point  of  the  surface  of  a  vessel,  is  normal  to 
the  surface,  and  is  proportional  to  the  depth  of  the  point 
below  the  free  surface.  Hence,  if  an  orifice  be  made  at 
any  point,  the  liquid  will  flow  out  in  a  jet,  normal  to  the 
surface  at  that  point,  and  with  a  velocity  due  to  the  dis- 
tance of  the  orifice  from  the  free  surface  of  the  fluid. 

If  the  orifice  is  on  a  vertical  side  of  a  vessel,  the  initial 
direction  of  the  jet  will  be  horizontal ;  if  it  be  at  a  point 
where  the  tangent  plane  is  oblique 
to  tlie  horizon,  the  initial  direc- 
tion of  the  jet  will  be  oblique  ;  if 
the  opening  is  on  the  upper  side 
of  a  portion  of  a  vessel  where  the  ^'^*  ■^^^• 

tangent  is  horizontal,  the  jet  will  be  directed  upward,  and 
will  rise  to  a  height  due  to  the  velocity;  that  is,  to  the 
height  of  the  upper  surface  of  the  fluid. 

Modification  due  to  Extraneous  Pressure. 

165.  If  the  upper  surface  of  the  liquid,  in  any  of  the 
preceding  cases,  be  pressed  by  a  force,  as  when  it  is  urged 

10* 


't^ 

Ij 

"nB 

5 

,-— V-.-. 

'V 

•,     1 

226  MECHANICS. 

downward  by  a  piston,  we  may  denote  the  height  of  a  col- 
umn of  the  fluid  whose  weight  is  equal  to  the  extraneous 
pressure,  by  li.  The  velocity  of  efflux  will  then  be  given 
by  the  equation, 

v=  ^/mii  +  h'). 

The  pressure  of  the  atmosphere  acts  equally  on  the 
upper  surface  and  the  opening ;  hence,  in  ordinary  cases, 
it  may  be  neglected ;  but  were  the  liquid  to  flow  into  a 
vacuum,  or  into  rarefied  air,  the  pressure  must  be  taken 
into  account,  and  this  may  be  done  by  means  of  the  for- 
mula just  given. 

Should  the  flow  take  place  into  condensed  air,  or  into 
any  medium  which  opposes  a  greater  resistance  than  the 
atmospheric  pressure,  the  extraneous  pressure  would  act 
upward,  h'  would  be  negative,  and  the  preceding  formula 
would  become. 


Spouting  of  Liquids  on  a  Horizontal  Plane. 

166.  Let  KL  be  a  vessel  filled  with  water,  D  an  orifice 
in  its  vertical  side,  and  DB  the  path  of  the  spouting  fluid. 
We  may  regard  each  drop  as  a  projectile 
shot  fortli  horizontally,  and  then  acted    ^b|"'""^> 
on  by  gravity.     Its  path  is,  therefore,  a    ^m     ^^^^\ 
parabola,  and  the  circumstances  of  its    ^Dp"i"—y\\ 
motion  are  made  known  by  equations    ^D — ^~"y^5;^"v\ 
(89)  and  (94).  ^^Iri" 

Denote  DK,  by  h',  and  DL,  by  h.  We  Fig.  m 

have,  from  equation   (94),  by  making  y  equal  to  h',  and 
x=KB,  


MECHANICS    OF    LIQUIDS.  227 

But  we  have  found  v  =  ^/'^(jh  ;  hence,  by  substitution, 

If  we  describe  a  semicircle  on  KL,  and  through  Z>-draw 
an  ordinate,  DH,  we  have,  from  a  property  of  the  circle, 

DH  =  ^DK  .  DL  =:  Vhh\ 

Hence  we  have,  by  substitution, 

JCB=2DH, 

Since  there  are  two  points  on  ICL  at  which  the  ordinates 
are  equal,  there  must  be  two  orifices  through  which  the 
fluid  will  spout  to  the  same  distance  on  the  horizontal 
plane ;  one  of  these  is  as  far  above  the  centre,  0,  as  the 
other  is  below  it. 

If  the  orifice  be  at  0,  midway  between  K  and  L,  the 
ordinate,  OS,  will  be  greatest  possible,  and  the  range,  KB', 
will  be  a  maximum.  The  range  in  this  case  will  be  equal 
to  the  diameter  of  the  circle,  LHK,  or  to  the  distance  from 
the  surface  of  the  water  in  the  vessel  to  the  horizontal 
plane. 

If  the  jet  is  directed  obliquely  upward  by  a  short  pipe, 
A,  (Fig.  138),  the  path  described  by  each  particle  will  still 
be  the  arc  of  a  parabola,  ABC.  Since  each  particle  of  the 
liquid  may  be  regarded  as  a  body  projected  obliquely  up- 
ward, the  nature  of  the  path  and  the  circumstances  of  the 
motion  will  be  given  by  equation  (89). 

If  a  semi-parabola,  LB',  is  described,  having  its  axis  ver- 
tical, its  vertex  at  L,  and  focus  at  K,  then  may  every  point, 
P,  within  the  curve,  be  reached  by  two  separate  jets  issuing 
from  the  side  of  the  vessel ;  every  point  on  the  curve  can 
be  reached  by  one,  and  only  one  ;  points  lying  without  the 
curve  cannot  be  reached  by  any  jet  whatever. 

In  like  manner,  the  same  equation  will  make  known  the 


228  MECHANICS. 

nature  of  the  p«ath  and  the  circumstances  of  motion,  when 
the  jet  is  directed  obliquely  downward  by  a  short  tube. 

Coefficients  of  Efflux  and  Velocity. 

167.  When  a  vessel  empties  itself  by  a  small  orifice  at  the 
bottom,  it  is  observed  that  the  particles  of  fluid  near  the  top 
descend  in  vertical  lines  ;  when  they  approach  the  bottom 
they  incline  toward  the  orifice,  the  converging  lines  of  par- 
ticles tending  to  cross  each  other  as  they  emerge  from  the 
vessel.  The  result  is,  the  stream  growls  narrower,  after 
leaving  the  vessel,  until  it  reaches  a  point  at  a  distance 
from  the  vessel  equal  to  about  the  radius  of  the  orifice, 
when  the  contraction  becomes  a  minimum,  and  below  that 
point  the  vein  again  spreads  out.  This  phenomenon  is 
called,  contraction  of  the  vein.  The  cross  section  at  the 
most  contracted  part  is  not  far  from  -^^  of  the  area  of  the 
orifice,  when  the  vessel  is  very  thin.  If  we  denote  the  area 
of  the  orifice,  by  a,  and  the  area  of  the  least  cross  section 
of  the  vein,  by  a,  we  have, 

a'  =  ha, 

in  which  h  is  a  number  to  be  determined  by  experiment. 
This  number  is  called  the  coefficient  of  contraction. 

To  find  the  quantity  of  water  discharged  through  an 
orifice  at  the  bottom  of  the  containing  vessel,  in  one 
second,  we  multiply  the  smallest  section  of  the  vein  by  the 
velocity.  Denoting  the  quantity  discharged  in  one  second, 
by  Q\  we  have, 

§'  =  ha  ^/^gh. 

This  formula  is  only  true  on  the  supposition  that  the 
actual  velocity  is  the  same  as  the  theoretical  velocity,  which 
is  not  the  case,  as  has  been  shown  by  experiment.     The 


MECHANICS    OF   LIQUIDS.  229 

theoretical  velocity  is  equal  to  ^/^gli,  and  if  we  denote  the 
actual  velocity,  by  v\  we  have, 

v'  =  I  \/lgh, 

in  which  I  is  to  be  determined  by  experiment;  this  value 
of  /is  slightly  less  than  1,  and  is  called  the  coefficient  cf 
velocity.  In  order  to  get  the  actual  discharge,  we  must  re- 
place V2^A  by  l^'2gli,  in  the  preceding  equation.  Doing 
so,  and  denoting  the  actual  discharge  per  second,  by  Q,  we 
have, 

Q  =  Ma  \/'2gh. 

The  product,  M,  is  called  the  coefficient  of  efflux.  It  has 
been  shown  by  experiment,  that  this  coefficient  for  orifices 
in  thin  plates,  is  not  quite  constant.  It  decreases  slightly, 
as  the  area  of  the  orifice  and  the  velocity  are  increased ; 
and  it  is  further  found  to  be  greater  for  circular  orifices 
than  for  those  of  any  other  shape. 

If  we  denote  the  coefficient  of  efflux,  by  in,  we  have, 

Q  =  via  \/2gh. 

In  this  equation,  h  is  called  head  of  water.  Hence,  we 
may  define  the  head  of  water  to  be  the  distance  from  the 
orifice  to  the  plane  of  the  upper  surface  of  the  fluid. 

The  mean  value  of  m  corresponding  to  orifices  of  from 
^  to  6  inches  in  diameter,  with  from  4  to  20  feet  head  of 
water,  has  been  found  to  be  about  .615.  If  w^e  take  k  =  .64, 
we  have, 

M  _  ^615^  _ 
^-  k  -  .640  -  ••^^• 

That  is,  the  actual  velocity  is  only  -j^^^  of  the  theoretical 
velocity.     This  diminution  is  due  to  friction,  viscosity,  &c. 


230  MECHANICS. 

Efflux  through  Short  Tubes. 

168.  It  is  found  that  the  discharge  from  a  given  orifice 
increases,  when  the  thickness  of  the  plate  through  which 
the  flow  takes  place  increases ;  also,  when  a  short  tube  is 
introduced. 

When  a  tube,  AB,  is  employed  not  more  than  four  times 
as  long  as  the  diameter  of  the  orifice,  the  value  of  m  be- 
comes, on  an  average,  equal  to  .813  ;  that 
is,  the  discharge  per  second  is  1.325  times 
as  great,  when  the  tube  is  used,  as  without 
it.  In  using  the  cylindrical  tube,  the  con- 
traction takes  place  at  the  outlet  of  the 
vessel,  and  not  at  the  outlet  of  the  tube. 

Compound  mouth-pieces  are   sometimes     I  [ 

formed  of   two  conic  frustums,   as   shown  '^v.:!/^ 

in  the  figure,  having  the  form  of  the  vein.  /ilT 

It  has  been  shown  by  Etelweiist,  that  the  ■';;;■.'.• 

most   effective  tubes  of  this  form   should  ?ig.  i4i. 

have  the  diameter,  CD,  equal  to  .833  of  AB.  The  angle 
made  by  the  sides,  CF  and  DE,  should  be  about  5°,  and 
the  length  of  this  portion  should  be  three  times  that  of  the 
other. 

Examples. 

1.  With  what  theoretical  velocity  will  water  issue  from  a  small 
orifice  16 1\  feet  below  the  surface  of  the  fluid  ?  Ans.  32^  ft. 

2.  If  the  area  of  the  orifice,  in  the  last  example,  is  -j^-  of  a  square 
foot,  and  the  coefiicient  of  efflux  .615,  how  many  cubic  feet  of  water 
will  be  discharged  per  minute?  Ans.  118.695  ft. 

3.  A  vessel,  constantly  filled  with  water,  is  4  feet  high,  with  a 
cross  section  of  one  square  foot ;  an  orifice  in  the  bottom  has  an  area 
of  one  square  inch.  In  what  time  will  three-fourths  of  the  water  be 
drawn  off,  the  coefficient  of  efflux  being  .6  ? 

Ans.  i  minute,  nearly. 

4.  A  vessel  is  kept  constantly  full  of  water.     How  many  cubic  feet 


MECHxVNICS   OF    LIQUIDS.  231 

■will  be  discharged  per  minute  from  an  orifice  9  feet  below  the  upper 
surface,  having  an  area  of  one  square  inch,  the  coefficient  of  efflux 
being  .6  V  Ans.  6  cubic  feet,  about. 

5.  In  the  last  example,  what  will  be  the  discharge  per  minute,  if 
we  suppose  each  square  foot  of  the  upper  surface  to  be  pressed  by  a 
force  of  645  lbs.  ?  Ans.  S'i  cubic  feet,  about. 

6.  The  head  of  water  is  16  feet,  and  the  orifice  is  too  of  a  square 
foot.  What  quantity  of  water  will  be  discharged  per  second,  when 
the  orillce  is  through  a  thin  plate? 

SOLUTION. 

In  this  ;ase,  we  have, 


Q  ^  .615  X  .01 V2  X  '62^  X  16  =  .197  cubic  feet 
When  a  shori  cylindrical  tube  is  used,  we  have, 
^  =  .197  X  1.325  =  .261  cubic  feet 

Capillary  Phenomena. 

169.  When  a  liqaid  is  in  equilibrium,  under  the  action 
of  its  own  weight,  it  has  been  shown  that  its  upper  surface 
is  level.  It  is  observed,  however,  in  the  neighborhood  of 
solid  bodies,  such  as  the  walls  of  a  vessel,  that  the  surface 
is  sometimes  elevated,  and  sometimes  depressed,  according 
to  the  nature  of  the  liquid  and  solid  in  contact.  These 
elevations  and  depressions  result  from  the  action  of  mo- 
lecular forces,  exerted  between  the  particles  of  the  liquid 
and  solid  in  contact;  from  the  fact  that  they  are  more 
apparent  in  small  tubes,  of  the  diameter  of  a  hair,  they 
have  been  called  cainllary  phenomena,  and  the  forces  giv- 
ing rise  to  them,  capillary  forces. 

The  following  are  some  of  the  observed  effects  of  capil- 
lary action  :  When  a  solid  is  plunged  into  a  liquid  capable 
of  moistening  it,  as  when  glass  is  plunged  into  water,  the 
surface  of  the  liquid  is  heaped  tip  about  the  solid,  taking 
a  concave  form,  as  shown  in  Fig.  142. 

When  a  solid  is  pi  tinged  into  a  liquid  not  capable  of 


232  MECHANICS. 

moistening  it,  as  when  glass  is  plunged  into  mercury,  the 
surface  of  the  liquid  is  depressed  about  the  < 

solid,  taking  a  convex  form,  as  shown  in  J 

Fig.  143.  ^"^ 

The  surface  of  the  liquid  in  the  neighbor-  Fig.  142. 

hood  of  the  surfaces  of  the  containing  ves- 
sel takes  the  concave  or  convex  form  ac- 
cording  as   the   material  of   the   vessel  is 
capable  of  being  moistened,  or  not,  by  the      -^ 
liquid. 

Fig,  143 

These  phenomena  become  more  a])parent 
when  we  plunge  a  tube  into  a  liquid;  according  as  the  tube 
is,  or  is  not,  capable  of  being  moistened  by  the  liquid,  the 
liquid  will  rise  in  the  tube,  or  be  depressed  in  it.  When 
the  liquid  rises  in  the  tube,  its  upper  surface  takes  a  con- 
cave shape ;  when  it  is  depressed,  it  takes  a  convex  form. 
The  elevations,  or  depressions,  increase  as  the  diameter  of 
the  tube  diminishes. 

Elevation  and  Depression  between  Plates. 

170.  If  two  plates  of  any  substance  be  placed  parallel  to 
each  other,  it  is  found  that  the  laws  of  ascent  and  descent 
of  the  liquid  into  Avhich  they  are  plunged,  are  the  same  as 
for  tubes.  For  example :  if  two  plates  of  glass  parallel  to 
each  other,  and  pretty  close  together,  are  plunged  into 
water,  it  is  found  that  the  water  Avill  rise  between  them  to 
a  height,  inversely  proportional  to  their  distance  apart; 
and  further,  that  this  height  is  equal  to  about  one-half 
the  height  to  which  water  would  rise  in  a  glass  tube  whose 
internal  diameter  is  equal  to  the  distance  between  the 
plates. 

If  the  same  plates  be  plunged  into  mercury,  there  will  be 
a  depression  according  to  a  corresponding  law. 


MECHAN^ICS   OF   LIQUIDS.  233 

If  two  plates  of  glass,  AB  and  AC,  inclined  to  each 
other,  as  shown  in  Fig.  144,  be  pliuiged  into  a  liquid  that 
will  moisten  them,  the  liquid  will  rise 
between  tliem.  It  will  rise  higher  near 
the  junction,  the  surface  taking  a 
curved  form,  such  that  any  section 
made  by  a  plane  through  AD,  will  be  ^*^"  ^^ 

an  equilateral  hyperbola. 

If  the  line  of  junction  of  the  two  plates  is  horizontal,  a 
small  quantity  of  a  liquid  that  will  moisten  ^^^^^^-^ 
them,  assumes  the  shape  shown  at  A  ;  if  it  do 
not  moisten  them,  it  takes  the  form  shown 
at^. 

Attraction  and  Repulsion  of  Floating  Bodies. 

171.  If  two  small  balls  of  wood,  both  of  which  can  be 
moistened  by  water,  or  two  small  balls  of  wax,  that  cannot 
be  moistened,  be  placed  in  a  vessel  of  water,  and  brought 
so  near  each  other  that  the  surfaces  of  capillary  elevation, 
or  depression  interfere,  the  balls  will  attract  each  other 
and  come  together.  If  one  ball  of  wood  and  one  of  wax 
be  brought  so  near  that  the  surfaces  of  capillary  elevation 
and  depression  interfere,  the  bodies  will  repel  each  other, 
and  S3parate.  If  two  needles  be  carefully  oiled  and  laid 
on  the  surface  of  water,  they  will  repel  the  water  from 
their  neigliborhood,  and  float.  If,  whilst  floating,  they 
are  brought  sufficiently  near  to  each  other  to  permit  the 
surfaces  of  capillary  depression  to  interfere,  the  needles 
will  immediately  rush  together.  The  reason  of  the  needles 
floating  IS,  that  they  repel  the  water,  heaping  it  up  on  each 
side,  thus  forming  a  cavity  in  the  surface;  the  needle  is 
buoyed  up  by  a  force  equal  to  the  weight  of  the  displaced 
flnid,  and,  when  this  exceeds  the  weight  of  the  needle,  it 


234  MECHAKICS. 

floats.  On  this  principle  certain  insects  move  freely  over 
a  sheet  of  water;  their  feet  are  lubricated  with  an  oily  sub- 
stance which  repels  the  water,  producing  a  hollow  around 
each  foot,  and  gives  rise  to  a  buoyant  effort  greater  than 
the  weight  of  the  insect. 

The  principle  of  mutual  attraction  between  bodies,  both 
of  which  repel  water,  or  both  of  which  attract  it,  accounts 
for  the  fact  that  small  floating  bodies  have  a  tendency  to 
collect  in  groups  about  the  borders  of  the  containing  ves- 
sel. When  the  material  of  which  the  vessel  is  made,  exer- 
cises a  different  capillary  action  from  that  of  the  floating 
particles,  they  will  aggregate  themselves  at  a  distance  from 
the  surface  of  the  vessel. . 

Applications  of  the  Principles  of  Capillarity. 

nt.  It  is  a  consequence  of  capillary  action  that  water 
rises  to  fill  the  pores  of  a  sponge,  or  lump  of  sugar.  The 
same  principle  causes  oil  to  rise  in  the  wick  of  a  lamp, 
which  is  but  a  bundle  of  fibres  very  nearly  in  contact, 
leaving  capillary  interstices  between  them. 

Tlie  siphon  filter  is  the  same,  in  principle,  as  the  wick  of 
a  lamp.  It  consists  of  a  bundle  of  fibres  like  a  lamp-wick, 
one  end  of  which  dips  into  the  liquid  to  be  filtered,  whilst 
the  other  hangs  over  the  edge  of  the  vessel.  The  liquid 
ascends  the  fibrous  mass  by  capillary  attraction,  and  con- 
tinues to  advance  till  it  reaches  the  overhanging  end,  when, 
if  this  is  lower  than  the  upper  surface  of  the  liquid,  it 
will  fall  by  drops  from  the  end  of  the  wick,  the  impurities 
being  left  behind. 

The  principle  of  capillary  attraction  is  used  for  splitting 
rocks  and  raising  weights.  To  employ  this  principle  in 
cleaving  mill-stones,  as  is  done  in  France,  the  stone  is  first 
dressed  to  the  form  of  a  cylinder  of  the  required  diameter. 


MECHANICS    OF    LIQUIDS.  235 

Grooves  are  then  cut  around  it  where  the  divisions  are  to 
take  place,  and  into  these  grooves  thoroughly  dried  wedges 
of  willow-wood  are  driven.  On  being  exposed  to  the 
action  of  moisture,  the  cells  of  the  wood  absorb  water, 
expand,  and  finally  split  the  rock. 

To  raise  a  weight,  a  thoroughly  dry  cord  is  fastened  to 
the  weight,  and  then  stretched  to  a  point  above.  If  the 
cord  be  moistened,  the  fibres  absorb  moisture,  expand 
laterally,  the  rope  is  diminished  in  length,  and  the  weight 
raised. 

The  principle  of  capillary  action  is  also  employed  in 
metallurgy,  in  purifying  metals,  by  cupellation. 

Endosmose  and  Exosmose. 

173.  The  names  e7iclosmose  and  exosmose  have  been 
given  to  currents,  flowing  in  contrary  directions  between 
two  liquids,  when  separated  by  a  porous  partition,  either 
organic  or  inorganic.  The  discovery  of  this  phenomena  is 
due  to  M.  DuTROCHET,  who  called  the  flowing  in,  endos- 
mose,  and  the  flowing  out,  exosmose.  The  existence  of 
the  currents  was  established  by  an  instrument,  to  which 
he  gave  the  name  endosmometer.  This  instrument  con- 
sists of  a  tube  of  glass,  at  one  end  of  which  is  attached  a 
membranous  sack,  secured  by  a  ligature.  If  the  sack  be 
filled  with  gum  water,  a  solution  of  sugar,  albumen,  or 
almost  any  solution  denser  than  water,  and  then  plunged 
into  water,  it  is  observed,  after  a  time,  that  the  fluid  rises 
in  the  stem,  and  is  depressed  in  the  vessel,  showing  that 
water  has  entered  the  sack  by  passing  through  the  pores. 
By  applying  suitable  tests,  it  is  also  found  that  a  portion 
of  the  liquid  in  the  sack  has  passed  through  the  pores 
into  the  vessel. 

Two  currents  are  thus  established.     If  the  operation  be 


236  MECHANICS. 

reversed,  and  tlie  bladder  and  tube  be  filled  with  pure 
water,  the  liquid  in  the  vessel  will  rise,  whilst  that  in  the 
tube  falls.  The  phenomena  of  endosmose  and  exosmose 
are  extremely  various,  and  serve  to  explain  a  great  variety 
of  interesting  facts  in  animal  and  vegetable  physiology. 
The  cause  of  the  currents,  is  the  action  of  molecular  forces 
between  the  particles  of  the  bodies  employed. 


CHAPTER  VIII. 

MECHANICS    OF    GASES    AND     VAPORS. 

Gases  and  Vapors. 

174.  Gases  and  vapors  are  distinguished  from  other 
fluids,  by  their  great  compressibility,  and  correspondingly 
great  elasticity.  These  fluids  continually  tend  to  occupy 
a  greater  space ;  this  expansion  goes  on  till  counteracted 
by  some  extraneous  force,  as  that  of  gravity,  or  the  resist- 
ance offered  by  a  containing  vessel. 

The  force  of  expansion,  common  to  gases  and  vapors,  is 
called  their  tension,  or  elastic  force.  We  shall  take  for  the 
unit  of  this  force,  at  any  point,  the  pressure  that  would 
be  exerted  on  a  square  inch,  were  the  pressure  the  same  at 
every  point  of  the  square  inch  as  at  the  point  in  question. 
If  we  denote  this  unit,  by  p,  the  area  pressed,  by  a,  and  the 
entire  pressure,  by  P,  we  have, 

r  :=ap (121) 

Most  of  the  principles  demonstrated  for  liquids  hold 
good  for  gases  and  vapors,  but  there  are  certain  properties 
arising  from  elasticity  that  are  peculiar  to  aeriform  fluids, 
some  of  which  it  is  now  proposed  to  investigate. 

Atmospheric  Air. 

175.  The  gaseous  fluid  that  envelops  our  globe,  and 
extends  on  all  sides  to  a  distance  of  many  miles,  is  called 
the  atmosjjhere.  It  consists  principally  of  nitrogen  and 
oxygen,  together  with  small  but  variable  portions  of  Avatery 


238  MECHANICS. 

vapor  and  carbonic  acid,  all  in  a  state  of  mixture.  On 
an  average,  it  is  found  that  1000  parts  by  volume  of 
atmospheric  air,  taken  near  the  surface  of  the  earth,  con- 
sist of  about, 

788  parts  of  nitrogen, 

197  parts  of  oxygen, 
14  parts  of  watery  vapor, 
1  part  of  carbonic  acid. 
The  atmosphere  may  be  taken  as  a  type  of  gases,  for  it 
is  found  that  the  laws  regulating  density,  expansibility, 
and  elasticity,  are  the  same  for  all  gases  and  vapors,  so 
long  as  they  maintain  a  purely  gaseous  form.     It  is  found, 
however,  in  the  case  of  vapors,  and  of  those  gases  which 
have  been  reduced  to  a  liquid  form,  that  the  law  changes 
just  before  actual  liquefaction. 

This  change  appears  somewhat  analogous  to  that 
observed  when  water  passes  from  the  liquid  to  the  solid 
form.  Although  water  does  not  actually  freeze  till 
reduced  to  a  temperature  of  32°  Fah.,  it  is  found  that 
it  reaches  its  maximum  density  at  about  39°,  at  which 
temperature  the  particles  seem  to  commence  arranging 
themselves  according  to  new  laws,  preparatory  to  taking 
the  solid  form. 


Atmospheric  Pressure.  (j 

176.  If  a  tube,  35  or  36  inches  long,  open  at  b 
one  end  and  closed  at  the  other,  be  filled  with 
pure  mercury,  and  inverted  in  a  basin  of  the  same, 
the  mercury  will  fall  in  the  tube  until  the  vertical 
distance  from  the  surface  of  the  mercury  in  the 
tube  to  that  in  the  basin  is  about  30  inches.  This 
column  of  mercury  is  sustained  by  the  pressure  of 
the  atmosphere  exerted  on  the  surface  of  the  mer- 


i 


Fig.  146. 


MECHANICS   OF   GASES    AND   VAPORS.  239 

cury  in  the  basin,  and  transmitted  through  the  fluid,  accord- 
ing to  the  general  law  of  transwhsiooi  of  pressures.  The 
column  of  mercury  sustained  by  the  elasticity  of  the 
atmosphere  is  called  the  harometric  cohimn,  because  it  is 
generally  measured  by  an  instrument  called  a  barometer. 
In  fact,  the  instrument  just  described,  when  provided 
with  a  suitable  scale  for  measuring  the  height  of  the 
column,  is  a  complete  barometer.  The  height  of  the  baro- 
metric column  fluctuates  somewhat,  even  at  the  same 
place,  on  account  of  changes  of  temperature,  and  other 
causes  yet  to  be  considered. 

Observation  has  shown,  that  the  average  height  of  the 
barometric  column  at  the  level  of  the  sea,  is  a  trifle  less 
than  30  inches. 

The  weight  of  a  column  of  mercury  30  inches  high, 
having  a  cross  section  of  one  inch,  is  nearly  15  pounds. 
Hence,  the  unit  of  atmospheric  pressure  is  15  pounds. 

This  unit  is  called  an  atmosphere,  and  is  often  employed 
in  estimating  the  pressure  of  elastic  fluids,  particularly 
steam.  Hence,  to  say  that  the  pressure  of  steam  in  a  boiler 
is  two  atmospheres,  is  equivalent  to  saying,  that  there  is  a 
pressure  of  30  pounds  on  each  square  inch  of  the  interior 
of  the  boiler.  In  general,  when  we  say  that  the  tension  of 
a  gas  or  vapor  is  7i  atmospheres,  we  mean  that  each  square 
inch  is  pressed  by  a  force  of  7i  times  15  pounds. 

Mariotte's  Law. 

177.  When  a  given  mass  of  gas  or  vapor  is  compressed 
so  as  to  occupy  a  smaller  space,  its  elastic  force  is  increased ; 
if  its  volume  is  increased,  its  elastic  force  is  diminished. 

The  law  of  increase  and  diminution  of  elastic  force,  dis- 
covered by  Mariotte,  and  bearing  his  name,  may  be 
enunciated  as  follows : 


240  MECHANICS. 

The  elastic  force  of  a  given  mass  of  gas,  whose  tem- 
perature rernains  the  same,  varies  inversely  as  the  volume 
it  occupies. 

As  long  as  the  mass  remains  the  same,  its  density  varies 
inversely  as  its  volume.     Hence, 

The  elastic  force  of  a  gas,  whose  temperature  remaiyis  the 
same,  varies  as  its  density ;  and  conversely,  its  density 
varies  as  its  elastic  force. 

Mariotte's  law  may  be  verified  for  atmospheric  air,  by 
an  instrument  called  Mariotte's  Tube.  This  is  a  tube, 
ABCD,  of  imiform  bore,  bent  so  that  its  two 
branches  are  parallel  to  each  other.  The  shorter 
branch,  AB,  is  closed  at  its  upper  extremity, 
whilst  the  longer  one  is  open.  Between  the  two 
branches,  and  attached  to  the  frame,  is  a  scale  of 
equal  parts. 

To  use  the  instrument,  place  it  in  a  vertical 
position,  and  pour  mercury  into  the  tube,  until 
it  just  cuts  off  communication  between  the  two 
branches.  The  mercury  will  then  stand  at  the 
same  level,  EC,  in  both  branches,  and  the  tension  '""  ^^^' 
of  the  air  in  AB,  will  be  exactly  equal  to  that  of  the  ex- 
ternal atmosphere.  If  an  additional  quantity  of  mercury 
be  poured  into  the  longer  branch,  the  air  in  the  shorter 
branch  will  be  compressed,  and  the  mercury  will  rise  in 
both  branches,  but  higher  in  the  longer,  than  in  the 
shorter  one.  Suppose  the  mercury  to  have  risen  in  the 
shorter  branch,  to  K,  and  in  the  longer  one,  to  P.  There 
will  be  an  equilibrium  in  the  mercury  lying  below  tiie 
horizontal  plane,  KK ;  there  will  also  be  an  equilibrium 
between  the  tension  of  the  air  in  AK,  and  the  forces  which 
give  rise  to  that  tension.  These  forces  are,  tlie  pressure  of 
the  external  atmosphere,  transmitted  through  the  mercury, 


MECHANICS    OF   GASES   AND   VAPORS.  241 

and  the  weight  of  a  column  of  mercury  whose  base  is  the 
cross  section  of  the  tube,  and  whose  altitude  is  PK.  If  we 
denote  the  height  of  the  column  of  mercury  sustained  by 
the  pressure  of  the  external  atmosphere,  by  h,  the  tension 
of  the  air  in  AK,  will  be  measured  by  the  weight  of  a 
column  of  mercury,  whose  base  is  the  cross  section  of  the 
tube,  and  whose  height  is  h  +  PK.  Since  the  weight  is 
proportional  to  the  height,  the  tension  of  the  confined  air 
is  proportional  to  A  +  PK. 

Now,  whatever  may  be  the  value  of  PK,  it  is  found  that, 

AK  :  AB  ::  h  :  h  +  PK; 
whence, 

.j._AB  .h 

^^"--hTPK- 

If  PK=  h,  we  have,  AK=iAB;  \iPK=  2h,  we  have, 

AK=  ^AB  ;  if  PK  =  nh,  n  being  any  positive  number, 

AB 
entire  or  fractional,  we  have,  AK= — .    This  formula, 

deduced  from  Mariotte's  law,  was  verified  by  Dulong  and 
Arago  for  all  values  of  ?i,  up  to  n  =  27.  The  law  may 
also  be  verified  when  the  pressure  is  less  than  an  atmos- 
phere, by  the  following  apparatus :  AK  is  a  tube  of  uni- 
form bore,  closed  at  its  upper  and  open  at  its  lower 
extremity;    CD  is    a   deep   cistern   of  mercury.  ^ 

The  tube,  AK,  is  either  graduated  into  equal 
parts,  commencing  at  A,  or  has  attached  to  it  a 
scale  of  brass  or  ivory. 

To  use  the  instrument,  pour  mercury  into  the 
tube  till  it  is  nearly  full ;  place  the  finger  over  the 
open  end,  invert  it  in  the  cistern,  and  depress  it 
till  the  mercury  stands  at  the  same  level  without 
and  within  the  tube,  and  suppose  the  surface  of 


the  mercury  in  this  case  to  be  at  B.      Then  will    Fig.  i48. 

11 


242  MECHANICS. 

the  tension  of  the  air  in  AB,  be  equal  to  that  of  the  ex- 
ternal atmosphere.  If  the  tube  be  raised  vertically,  the  air 
inAB  will  expand,  its  tension  will  diminish,  and  the  mer- 
cury will  fall  in  the  tube,  to  maintain  the  equilibrium. 
Suppose  the  level  of  the  mercury  in  the  tube,  to  have 
reached  IC.  In  this  position  of  the  instrument  the  tension 
of  the  air  in  AIT,  added  to  the  weight  of  the  column  of  mer- 
cury, J^B,  will  be  equal  to  the  tension  of  the  external  air. 
Now,  it  is  found,  whatever  may  be  the  value  of  KB,  that 

AK  :  AB  ::  h  :  h  -EK; 
whence, 

AB  .h 


AK= 


h  -  EK' 


If    EK^  \lu  we  have,  AK^  2AB  j  if  EK=^h,  we 

have,  AK=  SAB;  in  general,  if  EK  = -li,  we  have, 

AK^  AB{n  +  1). 

This  formula  has  been  verified,  for  all  values  of  n,  up  to 
n  =  111. 

It  is  a  law  of  Physics  that,  when  a  gas  is  suddenly  com- 
pressed, heat  is  evolved,  and  when  a  gas  is  suddenly 
expanded,  heat  is  absorbed ;  hence,  in  making  the  experi- 
ment, care  must  be  taken  that  the  temperature  be  kept 
uniform. 

More  recent  experiments  have  shown  that  Mariotte's  law 
is  not  strictly  true,  especially  for  high  tensions,  yet  its  vari- 
ation is  so  small  that  the  error  committed  in  regarding  it 
as  true  is  not  appreciable  in  any  practical  case. 

Gay  liussac's  Law. 

178.  If  the  volume  of  any  gas  or  vapor  remain  the  same, 
and  its  temperature  be  increased,  its  tei  sion  is  increased 


MECHANICS    OF    GASES    AND    VAPORS.  243 

also.  If  the  pressure  remain  the  same,  the  volume  of  the 
gas  increases  as  the  temperature  is  raised.  The  law  of  in- 
crease and  diminution,  as  deduced  by  Gay  Lussac,  whose 
name  it  bears,  may  be  enunciated  as  follows : 

In  a  given  mass  of  gas,  or  vapor,  if  the  volume  remain 
the  same,  the  te7isio7i  varies  as  the  temperature  ;  if  the  ten- 
sion remain  the  same,  the  volume  varies  as  the  temperature. 

According  to  Eegnault,  if  a  given  mass  of  air  be  heated 
from  32°  Fahrenheit  to  212°,  the  tension  remaining  con- 
stant, its  volume  will  be  increased  by  the  .3665th  part  of 
its  volume  at  32°.  Hence,  the  increase  for  each  degree  of 
temperature  is  the  .00204th  part  of  its  volume  at  32°.  If 
we  denote  the  volume  at  32°,  by  v,  and  the  volume  at  the 
temperature,  t',  by  v',  we  have, 

v'  =  ?;[l  +  . 00204(^-32)] (122) 

Solving  with  reference  to  v,  we  have, 

...  (123) 


1  +  .00204(^'  -  32)   •  '  • 

Formula  (123)  enables  us  to  compute  the  volume  of  a 
mass  of  air  at  32°,  when  we  know  its  volume  at  the  temper- 
ature, t',  the  pressure  remaining  constant. 

To  find  the  volume  at  the  temperature,  t",  we  have  sim- 
ply to  substitute  t"  for  t'  in  (122).  Denoting  this  volume 
by  v",  we  have, 

v"  =  v[l  +  .00204(if"  •  -  32)]. 
Substituting  for  v  its  value,  from  (llA),  we  get, 

,  1  4-  .00204(r  -  32)  ,,,,., 

^    ^  '  1  +  .00204(V  -  32) ^''^^ 

This  formula  enables  us  to  compute  the  volume  of  a 
mass  of  air,  at  a  temperature,  t",  when  we  know  its  volume 
at  the  temperature,  t' ;  and,  since  the  density  varies  in- 


244 


MECHANICS. 


Tersely  as  the  volume,  we  may  also,  by  means  of  the  same 
formula,  find  the  density  of  any  mass  of  air,  at  the  temper- 
ature, /",  when  we  have  given  its  density  at  the  temper- 
ature, t'. 

Manometers. 
179.  A  manometer  is  an  instrument  for  measuring  the 
tension  of  gases  and  vapors,  particularly  of  steam.     Two 
principal  varieties  of  manometers  are  used  for  measuring 
the  tension  of  steam,  the  open,  and  the  closed  manometer. 


The  open  Manometer. 

180.  The  open  manometer  consists  of  an  open  glass 
tube,  AB,  terminating  near  the  bottom  of  a  cistern,  EF. 
The  cistern  is  of  w  rough t-iron,  steam  tight, 
and  filled  with  mercury.  Its  dimensions 
are  such,  that  the  upper  surface  of  the  mer- 
cury will  not  be  materially  lowered,  when  a 
portion  of  the  mercury  is  forced  up  the  tube.  '^ 
BD  is  a  tube,  by  means  of  which,  steam  may 
be  admitted  from  the  boiler  to  the  surface 
of  the  mercury  in  the  cistern.  This  tube  is 
sometimes  filled  with  water,  through  which 
the  pressure  of  the  steam  is  transmitted  to  Fig.  149. 

the  mercury. 

To  graduate  the  instrument.  All  communication  with 
the  boiler  is  cut  oif,  by  closing  the  stop-cock,  B^  and  com- 
munication with  the  external  air  is  made  by  opening  the 
stop-cock,  D.  The  point  of  the  tube,  AB,  to  which  the 
mercury  rises,  is  noted,  and  a  distance  is  laid  off,  upward, 
from  this  point,  equal  to  what  the  barometric  column 
wants  of  30  inches,  and  the  point,  II,  thus  determined,  is 
marked  1.     This  point  will  be  very  near  the  surface  of  the 


MECHAKICS   OF   GASES   AND   VAPORS.  245 

mercury  in  the  cistern.  From  the  point,  H,  distances  of 
30,  60,  90,  &c.,  inches  are  laid  off  upward,  and  the  corre- 
sponding points  numbered  2,  3,  4,  &c.  These  divisions 
correspond  to  atmospheres,  and  may  be  subdivided  into 
tenths  and  hundredths. 

To  use  the  instrument,  the  stop-cock,  />,  is  closed,  and 
communication  made  with  the  boiler,  by  opening  the  stop- 
cock, E.  The  height  to  which  the  mercury  rises  in  the 
tube  indicates  the  tension  of  the  steam  in  the  boiler,  which 
may  l)e  read  from  the  scale  in  terms  of  atmospheres  and 
decimals  of  an  atmosphere.  If  the  pressure  in  pounds  is 
wished,  it  may  at  once  be  found  by  multiplying  the  read- 
ing of  the  instrument  by  15. 

The  principal  objection  to  this  kind  of  manometer  is  its 
want  of  portability,  and  the  great  length  of  tube  required, 
when  high  tensions  are  to  be  measured. 

The  closed  Manometer. 

181.  The  general  construction  of  the  closed  manometer 
is  the  same  as  that  of  the  open  manometer,  except  that 
the  tube,  AB,  is  closed  at  the  top.  The  air  confined  in 
the  tube,  is  compressed  in  the  same  way  as  in  Mariotte's 
tube. 

To  graduate  this  instrument.  We  determine  the  divi- 
sion, H,  as  before.  The  remaining  divisions  are  found  by 
applying  Mariotte's  law. 

Denote  the  distance  in  inches,  from  H  to  the  top  of  the 
tube,  by  I;  the  pressure  on  the  mercury,  in  atmospheres, 
by  72,  and  the  distance  in  inches,  from  H  to  the  upper 
surface  of  the  mercury  in  the  tube,  by  x. 

The  tension  of  the  air  in  the  tube  is  equal  to  that  on 
the  mercury  in  the  cistern,  diminished  by  the  weight  of  a 


Si46  MECHANICS. 

column  of  mercury  whose  altitude  is  x.     Hence,  in  atmos- 
pheres, it  is 

X 

The  bore  of  the  tube  being  uniform,  the  volume  of  the 
compressed  air  is  proportional  to  its  height.  When  the 
pressure  is  1  atmosphere,  the  height  is  I ;  when  the  pres- 

X 

sure  is  n  —  —  atmospheres,  the  height  is  I  —  x.     Hence, 

from  Mariotte's  law, 

1  '.  n  —  —   :\  I  —  X  :  L 
ijyj 

Whence,  by  reduction, 

x""  -  (80?i  +  l)x=z  -  :m{n  -  1). 
Solving,  with  respect  to  :/:,  we  liave, 


30n  +  / 

2 


|/-30^(«-i)  +  (?5!L±i): 


The  upper  sign  of  the  radical  is  not  used,  as  it  would 
give  a  value  for  x,  greater  than  I.  Taking  the  lower  sign, 
and  assuming  I  =  30  in.,  we  have. 


X  =  1671  +  15  -  V-  900(n  -  1)  +  {I5n  +  15)'. 

Making  7i  =  2,  3,  4,  &c.,  in  succession,  we  find  for  x,  the 
values,  11.46  in.,  17.58  in.,  20.92  in.,  &c.  These  distances 
being  set  off  from  H,  upward,  and  marked  2,  3,  4,  &c., 
indicate  atmospheres.  The  intermediate  spaces  may  be 
subdivided  by  the  same  formula. 

In  making  the  graduation,  we  have  supposed  the  tem- 
perature to  remain  the  same.  If,  however,  it  does  not 
remain  the  same,  the  reading  of  the  instrument  must  be 
corrected  by  a  table  computed  for  the  purpose. 

The  instrliment  is  used  in  the  same  manner  as  that 


MECHANICS  OF  GASES  AKD  VAPORS.  247 

already  described.     Neither  can  be   used  for  measuring 
tensions  less  than  1  atmosphere. 

The  Siphon  Gauge. 

182.  The  siphon  gauge  is  used  to  measure  tensions  of 
gases  and  vapors,  less  than  an  atmosphere.  It  consists  of 
a  tnhe,  ABC,  bent  so  that  its  two  branches  are 
parallel.  The  branch,  BC,  is  closed  at  the  top, 
and  filled  with  mercury,  which  is  retained  by  the 
pressure  of  the  atmosphere;  the  branch,  J ^,  is 
open  at  the  top.  If  the  air  be  rarefied  in  any 
manner,  or,  if  the  mouth  of  the  tube  be  exposed 
to  the  action  of  a  gas  whose  tension  is  sufficiently  PJg-iso. 
small,  the  mercury  will  no  longer  be  supported  in  BC,  but 
will  fall  in  it  and  rise  in  BA.  The  distance  between  the 
surfaces  of  the  mercury  in  the  two  branches,  given  by  a 
scale  between  them,  indicates  the  tension  of  the  gas.  If 
this  distance  is  expressed  in  inches,  the  tension  can  be 
found,  in  atmospheres,  by  dividing  by  30,  or,  in  pounds,  by 
dividing  by  2. 

The  Diving-Bell. 

183.  The  diving-bell  is  a  bell-shaped  vessel,  open  at  the 
bottom,  used  for  descending  into  the  water.  The  bell  is 
placed  with  its  mouth  horizontal,  and  let 
down  by  a  rope,  AB,  the  whole  apparatus 
being  sunk  by  weights  properly  adjusted. 
The  air  contained  in  the  bell  is  com- 
pressed by  the  pressure  of  the  water,  but 
its  increased  elasticity  prevents  the  water 
from  rising  to  the  top  of  the  bell,  which 

is  provided  with  seats  for  the   accommodation    of  those 
within    the   bell.      The   air,  constantly  contaminated   bv 


248  MECHANICS. 

breathing,  is  continually  replaced  by  fresh  air,  pumped  in 
through  a  tube,  FG.  Were  there  no  additional  air  intro- 
duced, the  volume  of  the  compressed  air,  at  any  depth, 
might  be  computed  by  Mariotte's  law.  The  unit  of  the 
compressing  force,  in  this  case,  is  the  weight  of  a  column 
of  water  whose  cross  section  is  a  square  inch,  and  whose 
height  is  the  distance  from  DC  to  the  surface  of  the  water. 

The  Barometer. 

184.  The  barometer  is  an  instrument  for  measuring  the 
pressure  of  the  atmosphere.  It  consists  of  a  glass  tube, 
hermetically  sealed  at  one  extremity,  filled  with  mercury, 
and  inverted  in  a  basin  of  that  fluid.  The  pressure  of  the 
air  is  indicated  by  the  height  of  mercury  it  supports. 

A  variety  of  forms  of  mercurial  barometers  have  been 
devised,  all  involving  the  same  mechanical  principle.  The 
most  important  of  these  are  the  sipho7i  and  the  cUterii 
iarmneters. 

The  Siphon  Barometer. 

185.  The  siphon  barometer  consists  of  a  tube,  CDE, 
bent  so  that  its  two  branches,  CD  and  DE,  are  ^c 
parallel  to  each  other.  A  scale  is  placed  between  lltA 
them,  and  attached  to  the  same  frame  with  the 
tube.  The  longer  branch,  CD,  is  hermetically 
sealed  at  the  top,  and  filled  with  mercury;  the 
shorter  one  is  open  to  the  air.  When  the  instru-  ii 
ment  is  placed  vertic^ly,  the  mercury  sinks  in  the  j;: 
longer  branch  and  rises  in  the  shorter  one.  The  j) 
distance  between  the  surface  of  the  mercury  in  fU'.  152, 
the  two  branches  indicates  the  pressure  of  the  atmos- 
phere. 


MECHAI^ICS    OF   GASES   AND    VAPORS. 


249 


The  Cistern  Barometer. 


i 


OK 


186.  The  cistern  barometer  consists  of  a  glass  tube,  filled 
and  inverted  in  a  cistern  of  mercury.  The  tube  is  sur- 
rounded by  a  frame  of  metal,  attached  to  the  cistern.  Two 
longitudinal  openings,  near  the  upper  part  of  the  frame, 
permit  the  upper  surface  of  the  mercury  to  be  seen.  A 
slide,  moved  up  and  down  by  a  rack  and  pinion,  may  be 
brought  exactly  to  the  upper  level  of  the  mercury.  The 
height  of  the  column  is  then  read  from  a  scale, 
whose  0  is  at  the  surface  of  the  mercury  in  the 
cistern.  The  scale  is  graduated  to  inches  and 
tenths,  and  the  smaller  divisions  are  read  by  a 
vernier. 

The  figure  shows  the  parts  of  a  complete 
cistern  barometer;  KK  represents  the  frame; 
HH,  the  cistern,  of  glass,  at  the  upper  part, 
that  the  mercury  in  the  cistern  may  be  seen 
through  it;  L,  a  thermometer,  to  show  the  tem- 
perature of  the  mercury;  N^  the  sliding-ring 
bearing  the  vernier,  and  moved  up  and  down  by 
the  screw,  M. 

The  cistern  is  shown  on  an  enlarged  scale  in 
Fig.  154 ;  ^  is  the  barometer  tube,  terminating 
in  a  small  opening,  to  prevent  sudden .  shocks 
when  the  instrument  is  moved  from  place  to 
place;  H,  the  frame  of  the  cistern;  B,  the 
upper  portion  of  the  cistern,  made  of  glass, 
that  the  mercury  may  be  seen ;  E,  a  piece  of 
ivory,  projecting  from  the  upper  surface  of 
the  cistern,  whose  point  corresponds  to  the  0 
of  the  scale :  CC,  the  lower  part  of  the  cistern, 
made  of  leather,  or  other  flexible  material,  and 

11* 


D 

Ig.  153. 


3 


^ 


T 


D 

Fig.  154. 


250  MECHANICS. 

attached  to  the  glass  part ;  D,  a  screw,  working  through 
the  frame,  and  against  the  bottom  of  the  bag,  CC,  by  means 
of  a  plate,  P.  The  screw,  D,  serves  to  bring  the  surface 
of  the  mercury  to  the  point  of  ivory,  E,  and  also  to  force 
the  mercury  to  the  top  of  the  tube,  when  it  is  desired  to 
transport  the  barometer  from  place  to  place. 

To  use  this  barometer,  it  is  suspended  vertically,  and 
the  level  of  the  mercury  in  the  cistern  brought  to  the  point 
of  ivory,  E,  by  the  screw,  i);  a  smart  rap  on  the  frame 
will  detach  the  mercury  from  the  glass,  to  which  it  tends  to 
adhere.  The  ring,  N,  is  run  up  or  down  till  its  lower  edge 
appears  tangent  to  the  surface  of  the  mercury  in  the  tube, 
and  the  altitude  is  read  from  the  scale.  The  height  of  the 
attached  thermometer  should  also  be  noted. 

The  requirements  of  a  good  barometer  are,  sufficient 
width  of  tube,  perfect  purity  of  mercury,  and  a  scale  with 
an  accurately  graduated  vernier. 

The  bore  of  the  tube  should  be  as  large  as  practicable,  to 
diminish  the  eftect  of  capillary  action.  On  account  of  the 
repulsion  between  the  glass  and  mercury,  the  latter  is  de- 
pressed in  the  tube,  and  this  depression  increases  as  the 
diameter  of  the  tube  diminishes. 

In  all  cases,  this  depression  should  be  allowed  for,  and 
the  reading  corrected  by  a  table  computed  for  the  purpose. 

To  secure  purity  of  the  mercury,  it  should  be  carefully 
distilled,  and  after  the  tube  is  filled,  it  should  be  boiled  to 
drive  off  any  bubbles  of  air  that  might  adhere  to  the  tube. 

Uses  of  the  Barometer. 

187.  The  primary  object  of  the  barometer  is,  to  measure 
the  pressure  of  the  atmosphere.  It  is  used  by  mariners  as 
a  weather-glass.  It  is  also  employed  for  determming  the 
heights  of  points  on  the  earth's  surface,  above  the  level  of 


MECHANICS   OF   GASES   AND   VAPORS.  251 

the  ocean.  The  principle  on  which  it  is  employed  for  the 
latter  purpose  is,  that  the  pressure  of  the  atmosphere  at 
any  place  depends  on  the  weight  of  a  column  of  air  reach- 
ing from  the  place  to  the  upper  limit  of  the  atmosphere. 
As  we  ascend  above  the  level  of  the  ocean,  the  weight  of 
the  column  diminishes;  consequently,  the  pressure  be- 
comes less,  a  fact  that  is  shown  by  the  mercury  falling  in 
the  tube. 

Difference  of  Level, 

188.  Let  aB  represent  a  vertical  prism  of  air,  whose 
cross  section  is  one  square  inch.  Denote  the  pressure  at 
B,  bj  p,  and  at  aa',  by  p'j  denote  the  density  of  ^ 


the  air  at  B,  by  d,  and  at  aa'  by  d',  and  suppose 
the  temperature  throughout  the  column  to  be 
32°  Fah. 

Pass  a  horizontal  plane,  bb',  infinitely  near  to 
aa',  and  denote  the  weight  of  the  air  in  ab,  by  w. 
Conceive  the  entire  column  to  be  divided  by  hor-    Fig.  155. 
izontal  planes  into  prisms,  whose  weights  are  each  equal 
to  tv,  and  denote  their  heights,  beginning  at  a,  by  s,  s',  s", 
&c. 

From  Mariotte's  law,  we  have, 

p'~d'^        •'•    l!~~Jd^ 

The  air  throughout  each  elementary  prism  may  be 
regarded  as  homogeneous;  the  density  of  the  air  in 
ab  is  therefore  equal  to  its  weight,  divided  by  its  vol- 
ume into  gravity  (Art.  15).  But  its  volume  is  equal  to 
1  X  1  X  5  =  5  ;  hence, 

,,       10  w        1 

gs  g       d' 


252  MECHANICS. 


Substituting  for  — ,,  its  value  in  the  preceding  equation, 

we  have,  s  —  -^  X% (125) 

dg      p' 

From  the  formula  for  log.  (1  +  y),  deduced  in  algebra,  we 
have,  by  substituting  for  y  the  fraction  — ,  the  equation. 


(w  \       w        w        ,  w  0 

p  /       p        2p"       3p" 


w  .    .        . 
But  — :  is  infinitely  small ;  hence,  all  the  terms  in  the  second 

p'  J  ^  ' 

member,  after  the  first,  may  be  neglected,  giving, 

/       V       //  P        \     P     I 

w 
or,  finally,  —  =  l{p'  +  to)  —  lp\ 

in  which  I  denotes  the  Naperian  logarithm. 

In  this  equation,  p'  is  the  pressure  on  the  prism,  ah ; 
hence,  p^  +  w  is  the  pressure  on  the  next  prism  below,  that 
is,  on  be. 

w 
If  we  substitute  the  value  of  —  in  equation   (125),  we 

have,  for  the  height  of  aby 

Substituting  in  succession  for  j)',  the  values,  p'-\-  ic,  p'  4-  2w, 
p'  +  3w,  &c.,  we  find  the  heights  of  be,  cd,  &c.,  to  the  nt\i 
at  the  base,  B,  as  follows : 


MECHANICS   OF   GASES   AND   VAPORS.  253 


'"=%\-^^P'  +  ^'')-^^P'^^'')^^ 


,(«-!)' 


^\l{p'  +  n7v)  —  l{p'  +  {n  —  l)w)]. 


Adding  the  equations  member  to  member,  and  denoting 
the  sum  of  the  first  members,  which  will  be  equal  to  A  B, 
by  z,  we  have, 

But  7iw  is  the  weight  of  the  air  in  aB ;  hence,  we  have, 
p'  +  niv  =  p,  or, 

z  =  ^l^ (126) 

dg  p 

Denoting  the  modulus  of  the  common  system  of  loga- 
rithms by  i¥,  and  designating  common  logarithms  by  the 
symbol  log,  we  have, 

P     1      P 


Mdg     ^  p' 

The  pressures,  j5  andjf?',  are  measured  by  the  heights  of 
the  columns  of  mercury  which  they  will  support ;  denoting 
these  heights  by  H  and  H\  we  have, 

p'~H'' 
whence,  by  substitution, 

^-ik'^'w (1^^) 

We  supposed  the  temperature,  of  both  air  and  mercury, 
to  be  32°.  "J'o  make  the  formula  general,  let  T  be  the 
temperature  of  the  mercury  at  B,  T'  its  temperature  at  a, 
and  denote  the  corresponding  heights  of  the  barometric 


254  MECHANICS. 

column,  by  h  and  h' ;  also,  let  t  be  the  temperature  of  the 
air  at  B,  and  t'  its  temperature  at  a. 

P 
The  quantity^  is  the  ratio  of  the  density  of  the  air  at 

By  to  the  corresponding  pressure,  the  temperature  being 
32°.  According  to  Mariotte's  law,  this  ratio  remains 
constant,  whatever  may  be  the  altitude  of  B  above  the 
level  of  the  ocean. 

If  we  denote  the  latitude  of  the  place,  by  /,  we  have, 
(Art.  117), 

g=G{l  -{-  .005133sin='0. 

It  has  been  shown,  by  experiment,  that,  a  column  of 
mercury  when  heated,  increases  in  length  at  the  rate 
of -j^oth  of  its  length  at  32°,  for  .each  degree.     Hence, 

7.       Tr(^  a   T-^2\        ^9990  +  ^-32 
^  =^(^  +  ^99^;  =  ^ 9990 ' 

h=H[l  +  -^^^J=H  -— . 

Dividing  the  first  equation  by  the  second,  member  by 
member,  we  have, 

A-j^    9990+  r- 32 
h'~  H'  '  9990  +  r'  -  32  • 

IS 

Dividing  both  terms  of  the  coefficient  of  -jp  by  the  de- 
nominator, and  neglecting  T'—  32,  in  comparison  with 
9990,  we  have, 

h-  —d  4-  ——]=  ^  n  +  .0001  (T—  T)] 

Whence,  by  reduction, 

H       h  1 


H'~  h''l  -H  .0001(7^-  T')' 


MECHANICS    OF   GASES    AND   VAPORS.  255 

The  quantity  z  denotes,  not  only  the  height,  but  also  the 
volume  of  the  column  of  air,  aB,  at  32°.  When  the  tem- 
perature is  changed  from  32°,  the  pressures  remaining  the 
same,  this  volume  will  vary,  according  to  the  law  of  Gay 

LUSSAC. 

If  we  suppose  the  temperature  of  the  entire  column  to 
be  a  mean  between  the  temperatures  at  B  and  a,  which  we 
may  do  without  sensible  error,  the  height  of  the  column 
will  become,  equation  (122), 

z[l  +  .00204  ^^-^'  -  32)]  =41  +  .00102(^5  ^t'  -  64)]. 

Hence,  to  adapt  equation  (127)  to  the  conditions  pro- 
posed, we  must  multiply  the  value  of  z  by  the  factor, 

1  +  .00102(if  -\-r  -  64). 

TT 

Substituting  in  equation  (127),  for  -yj-,  and  ^,  the  values 

given  above,  and  multiplying  the  resulting  value  of  z,  by 
the  factor  1  +  .00102(^  -\-  t'  -  64),  we  have, 

/;    1  +  .00102(if  +  r  -  64)  ^^  h 


Md'    G{1  +  .005133sin'Z)       ^  h'[l  +  MOl{T-  T')] 

(128) 

The  factor  ,-7^  is  constant,  and  may  be  determined  as 
MdG  ^ 

follows :  Select  two  points,  one  considerably  higher  than 
the  other,  and  determine,  by  trigonometrical  measurement, 
their  difference  of  level.  At  the  lower  point,  take  the 
reading  of  the  barometer,  of  its  attached  thermometer,  and 
of  a  detached  thermometer  exposed  to  the  air.  Make  sim- 
ilar observations  at  the  upper  station.  These  observations, 
together  with  the  latitude  of  the  place,  will  give  all  the 
quantities  entering  equation  (128),  except  the  factor  in 
question.     Hence,  this  factor  may  be  deduced.     It  is  found 


2bij  MECHANICS. 

to  be  60345.51  ft.     Hence,  we  have,  finally,  the  barometric 
formula, 

z  =  60345.51  ft  X 

1  4-  .00102  it  -\-t'  -  64)  ,  h 

—  log 


1  +  .005133sin7  ^  h'[l  +  .0001(r  -  T')] .  .  (129) 

To  use  this  formula  for  determining  the  difference  of  level 
between  two  stations,  observe,  simultaneously,  if  possible, 
the  heights  of  the  barometer,  and  of  the  attached,  and  de- 
tached thermometers,  at  the  two  stations.  Substitute 
these  results  for  the  corresponding  quantities  in  the  for- 
mula; also  substitute  for  /  the  latitude  of  the  place,  and 
the  resulting  value  of  z  will  be  the  difference  of  level 
required. 

If  the  observations  cannot  be  made  simultaneously  at 
the  two  stations,  make  a  set  of  observations  at  the  lower 
station  ;  after  a  certain  interval,  make  a  set  at  the  upper 
station;  then,  after  an  equal  interval,  make  another  set  at 
the  lower  station.  Take  a  mean  of  the  results  of  obser- 
vation at  the  lower  station,  as  a  single  set,  and  proceed  as 
before. 

For  the  more  convenient  application  of  the  formula, 
tables  have  been  computed,  by  which  the  arithmetical 
operations  are  much  facilitated. 

Steam. 

189.  If  water  be  exposed  to  the  atmosphere,  at  ordinary 
temperatures,  a  portion  is  converted  into  vapor,  mixes  with 
the  atmosphere,  and  constitutes  one  of  the  elements  of  the 
aerial  ocean.  The  tension  of  watery  vapor  thus  formed,  is 
very  slight,  and  the  atmosphere  soon  ceases  to  absorb  any 
more.  If  the  temperature  of  the  water  be  raised,  an  addi- 
tional amount  of  vapor  is  evolved,  and  of  greater  tension. 


MECHANICS    OF   GASES    AND    VAPORS.  257 

When  the  temperature  is  raised  to  a  point  at  which  the  ten- 
sion of  the  vapor  is  equal  to  that  of  the  atmosphere,  ebulli- 
tion commences,  and  vaporization  goes  on  with  great 
rapidity.  If  heat  be  added  beyond  the  point  of  ebullition, 
neither  the  water,  nor  the  vapor  will  increase  in  tempera- 
ture till  all  the  water  is  converted  into  steam.  When 
the  barometer  stands  at  30  inches,  the  boiling-point  of  pure 
water  is  212°  Fah. 

If  we  take  the  quantity  of  heat  that  is  necessary  to  raise 
one  pound  of  water  from  the  temperature  32°  F.  to  the 
temperature  33°  F.,  as  a  unit  of  heat,  the  total  amount  of 
heat  necessary  to  raise  a  pound  of  water  from  32°  F.  to 
212°  F.  will  be  180  ^inits,  and  Regnault  has  shown  that 
the  additional  amount  of  heat  necessary  to  convert  the 
entire  pound  of  water  into  steam  of  the  temperature  212° 
F.  is  equal  to  966.6  units.  Hence,  we  say  that  it  requires 
180  +  966.6  or  1146.6  units  of  heat  to  convert  a  pound  of 
water  at  32°  F.  into  a  pound  of  steam  at  212°  F.  Of  this 
amount  966.6  units  are  said  to  become  latent,  that  is,  this 
amount  of  heat  is  employed  in  converting  the  water  at  212° 
into  steam  of  the  same  temperature.  From  this  we  see 
that  the  amount  of  heat  that  becomes  latent  in  converting 
a  quantity  of  water  at  212°  F.  into  steam  at  the  same  tem- 
perature, is  nearly  5^  times  as  much  as  is  required  to  raise 
it  from  the  temperature  32°  F.  to  the  boiling  point. 

If  steam  is  generated  under  a  pressure  greater  or  less 
than  one  atmosphere,  the  boiling  point  of  the  water  will 
be  either  greater  or  less  than  212°  F.  In  this  case, 
Regnault  has  shown  by  experiment  that  the  total  num- 
ber of  units  of  heat  required  to  convert  a  pound  of 
water  at  32°  F.  into  steam,  will  be  given  by  the 
formula, 

Q  =  1091.7  +  .305(2^-32°), 


S58  MECHANICS. 

in  which  t  is  the  boiling  point  of  water  under  the  given 
pressure  expressed  in  degrees  of  Fahrenheit's  scale. 

Thus,  to  convert  1  lb.  of  water  at  32°  F.  into  steam  of 
the  temperature  250°  F.,  would  require  1158.2  units 
of  heat. 

When  water  is  converted  into  steam  under  a  pressure  of 
one  atmosphere,  each  cubic  inch  produces  about  1700  cubic 
inches  of  steam,  of  the  temperature  of  212°;  or,  since  a 
cubic  foot  contains  1728  cubic  inches,  we  may  say,  in 
round  numbers,  that  a  cubic  inch  of  water  gives  a  cubic 
foot  of  steam. 

If  water  be  converted  into  steam  under  greater  or  less 
pressure  than  an  atmosphere,  the  density  is  increased  or 
diminished,  and,  consequently,  the  volume  is  diminished, 
or  increased.  The  temperature  being  also  increased  or 
diminished,  the  increase  of  density,  or  decrease  of  volume 
will  not  be  exactly  proportional  to  the  increase  of  pressure ; 
but,  for  purposes  of  approximation,  we  may  consider  the 
densities  as  directly,  and  the  volumes  as  inversely  propor- 
tional to  the  pressures  under  which  the  steam  is  generated. 
On  this  hypothesis,  if  a  cubic  inch  of  water  be  evaporated 
under  a  pressure  of  half  an  atmosphere,  it  will  afford  two 
cubic  feet  of  steam ;  if  generated  under  a  pressure  of  two 
atmospheres,  it  will  only  afford  half  a  cubic  foot  of  steam. 

Work  of  Steam. 

190.  When  water  is  converted  into  steam,  a  certain 
amount  of  work  is  generated,  and,  from  what  has  been 
shown,  this  work  is  very  nearly  the  same,  whatever  may  be 
the  temperature  at  which  the  water  is  evaporated. 

Suppose  a  cylinder,  whose  cross  section  is  one  square 
inch,  to  contain  a  cubic  inch  of  water,  above  which  is  an 


MECHANICS   OF   GASES   AND   VAPORS.  259 

air-tight  piston,  that  may  be  loaded  with  weiglits  at 
pleasure.  In  the  first  place,  if  the  piston  is  pressed  down 
by  a  weight  of  15  pounds,  and  the  inch  of  water  converted 
into  steam,  the  weight  will  be  raised  to  the  height  of 
1728  inches,  or  144  feet.  Hence,  the  quantity  of  work  is 
144  X  15,  or,  2160  units.  Again,  if  the  piston  be  loaded 
with  a  weight  of  30  pounds,  the  conversion  of  water  into 
steam  will  give  but  864  cubic  inches,  and  the  weight 
will  be  raised  through  72  feet.  In  this  case,  the  quan- 
tity of  work  will  be  72  X  30,  or,  2160  units,  as  before. 
"We  conclude,  therefore,  that  the  quantity  of  work  is  the 
same,  or  nearly  so,  whatever  may  be  the  pressure  under 
which  steam  is  generated.  We  also  conclude,  that  the 
quantity  of  work  is  nearly  proportional  to  the  amount  of 
fuel  consumed. 

Besides  the  quantity  of  work  developed  by  simply  con- 
verting water  into  steam,  a  further  quantity  of  work  is 
developed  by  allowing  the  steam  to  expand  after  entering 
the  cylinder.  This  principle  is  used  in  steam-engines 
working  expansively. 

To  find  the  quantity  of  work  developed  by  steam  acting 
expansively :  Let  AB  represent  a  cylinder,  closed  at  A,  and 
having  an  air-tight  piston,  B.  Suppose  the  steam 
to  enter  at  the  bottom  of  the  cylinder,  and  to  push 
the  piston  upward  to  (7,  and  then  suppose  the  ^ 
opening  at  which  the  steam  enters,  to  be  closed : 
if  the  piston  is  not  too  heavily  loaded,  the  steam 
will  continue  to  expand,  and  the  piston  will  be 
raised  to  some  position,  B.  The  expansive  force  ^^^'  ^'^^' 
of  the  steam  will  obey  Mariotte's  law,  and  the  quantity 
of  work  due  to  expansion  may  be  computed  by  the  formula 
in  the  next  article. 


UJ 


r~  ~i 

:::   ::: 

rf 

G    p    ^"^^ 

200  MECHANICS. 


Work  due  to  the  Expansion  of  a  Gas  or  Vapor. 

191.  Let  the  gas,  or  vapor,  be  confined  in  a  cylinder 
closed  at  its  lower  end,  and  having  a  piston  working  air- 
tight. When  the  gas  occupies  a  por- 
tion of  the  cylinder  whose  height  is  h, 
denote  the  pressure  on  each  square 
inch  of  the  piston,  by  p;  when  the 
gas  expands,  so  that  the  altitude  of  the 
column  becomes  x,  denote  the  pressure 
on  a  square  inch,  by  ^.  AC 

Since  the  volumes  of  the  gas,  under  ^'^''  ^^^' 

these  suppositions,  are  proportional  to  their  altitudes  we 
shall  have,  from  Mariotte's  law, 

p  \  y  '.'.  X  :  h; 
whence, 

xy  —  ph. 

If  p  and  h  are  constant,  and  x  and  y  vary,  the  above 
equation  will  be  that  of  an  equilateral  hyperbola  referred 
to  its  asymptotes. 

Draw  AC  perpendicular  to  AM,  and  on  these  lines,  as 
asymptotes,  construct  the  curve,  NLH,  from  the  equation, 
xy  —ph.  Make  AG  =  h,  and  draw  GH  parallel  to  AC; 
it  will  represent  the  pressure,  p.  Make  AM  —  x,  and  draw 
MJV  parallel  to  A  C;  it  will  represent  the  pressure,  y.  In 
like  manner,  the  pressure  at  any  heio-ht  of  the  piston  may 
be  constructed. 

Let  JCL  be  drawn  infinitely  near  to  GH,  and  parallel 
with  it.  The  elementary  area,  GKLH,  will  not  differ  sen- 
sibly from  a  rectangle  whose  base  is  p,  and  altitude  is  GK. 
Hence,  its  area  may  be  taken  as  »the  measure  of  the  Avork 
whilst  the   piston   is   rising  through  the  infinitely  small 


MECHANICS   OF   GASES   AND   VAPORS.  261 

space,  GIC.  In  like  manner,  the  area  of  any  infinitely 
small  element,  bounded  by  lines  parallel  to  A  C\  may  be 
taken  to  represent  the  work  whilst  the  piston  is  rising 
through  the  height  of  the  element.  If  we  take  the  sum 
of  all  the  elements  between  GH  and  JfjV,  this  sum,  or  the 
area,  GMNH,  will  represent  the  work  of  the  force  of 
expansion  whilst  the  piston  is  rising  from  G  to  M.  But 
the  area  included  between  an  equilateral  hyperbola  and 
one  of  its  asymptotes,  limited  by  lines  parallel  to  the  other 
asymptote,  is  equal  to  the  product  of  the  co-ordinates  of 
any  point,  multiplied  by  the  Naperian  logarithm  of  the 
quotient  obtained  by  dividing  one  of  the  limiting  ordinates 
by  the  other ;   or,  in   this   particular  case,  it  is  equal   to 

2)h  X  MI-     Hence,  if  we  designate  the  quantity  of  work 

performed  by  the  expansive  force  whilst  the  piston  is  mov- 
ing over  GMy  by  q,  we  shall  have. 

This  is  the  work  exerted  on  each  square  inch  of  the  piston ; 
if  we  denote  the  area  of  the  piston,  by  A,  and  the  total 
quantity  of  work,  by  Q,  we  shall  have, 

Q  =  Aph  X  ?(J)  =  Aphxl{j^ (130) 


CHAPTER  IX. 

HYDRAULIC    AND    PNEUMATIC    MACHINES. 
Definitions. 

192.  Hydraulic  machines  are  machines  for  raising 
and  distributing  water,  as  inimps,  siphons,  hydraulic  ranis, 
dec.  The  name  is  also  applied  to  machines  in  which  water 
power  is  the  motor,  or  in  which  water  is  employed  to 
transmit  pressures,  as  water -wlieels,  hydraulic  presses,  <^c. 

Pneumatic  machines  are  machines  to  rarefy  and  con- 
dense air,  or  to  impart  motion  to  air,  as  air-pumps,  venti- 
lating-blowers,  Sc.  The  name  is  also  applied  to  those 
machines  in  which  the  living  force  of  air  is  the  motive 
power,  such  as  windmills,  &c. 

Water    Pumps. 

193.  A  water  pump  is  a  machine  for  raising  water  from 
a  lower  to  a  higher  level,  by  the  aid  of  atmospheric  pressure. 
Three  separate  principles  are  employed  in  pumps :  the 
sucking,  the  lifting,  and  the  forcing  principle.  Pumps 
are  named  according  to  the  principle  employed. 

Sucking  and    Lifting   Pump. 

194.  This  pump  consists  of  a  barrel.  A,  to  the  lower 
extremity  of  which  is  attached  a  sucking-pipe,  B,  leading 
to  a  reservoir.  An  air-tight  piston,  C,  is  worked  up  and 
down  in  the  barrel  by  a  lever,  E,  attached  to  a  piston-rod, 
D ;  P  is  a  valve  opening  upward,  which,  when  the  pump  is 


HYDRAULIC    AND    PNEUMATIC    MACHINES. 


263 


Ca 


M_ 


Fig.  158. 


at  rest,  closes  by  its  own  weight.  This  valve  is  called  the 
pisfoti-valve.  A  second  valve,  G,  also  opening  upward,  is 
placed  at  the  junction  of  the  pipe  with 
the  barrel ;  this  is  called  the  sleeping- 
valve.  The  space,  LM,  through  which 
the  piston  moves  up  and  down,  is  the 
play  of  the  pisto?i. 

To  explain  the  action  of  the  pump ; 
suppose  the  piston  to  be  in  its  lowest 
position,  and  everything  in  equilibrium. 
If  the  extremity  of  the  lever,  U,  be  de- 
pressed, and  the  piston  raised,  the  air 
in  the  lower  part  of  the  barrel  is  rarefied,  and  that  in  the 
pipe,  B,  by  virtue  of  its  greater  tension,  opens  the  valve, 
and  a  portion  escapes  into  the  barrel.  The  air  in  the  pipe, 
thus  rarefied,  exerts  less  pressure  on  the  water  in  the 
reservoir  than  the  external  air,  and,  consequently,  the 
water  rises  in  the  pipe,  until  the  tension  of  the  internal  air, 
plus  the  weight  of  the  column  of  water  raised,  is  equal  to 
the  tension  of  the  external  air ;  the  valve,  G,  then  closes 
by  its  own  weight. 

If  the  piston  be  again  depressed  to  the  lowest  limit,  the 
air  in  the  lower  part  of  the  barrel  is  compressed,  its  tension 
becomes  greater  than  that  of  the  external  air,  the  valve,  P 
is  forced  open,  and  a  portion  of  the  air  escapes.  If  the 
piston  be  raised  once  more,  the  water,  for  the  same  reason 
as  before,  rises  still  higher  in  the  pipe,  and  after  a  few 
double  strokes  of  the  piston,  the  air  is  completely  exhausted 
from  beneath  the  piston,  the  water  passes  througli  the 
piston-valve,  and  finally  escapes  at  the  spout,  F. 

The  water  is  raised  to  the  piston  by  the  pressure  of  the 
air  on  the  surface  of  the  water  in  the  reservoir  ;  hence,  the 
piston  should  not  be  placed  at  a  greater  distance  above  the 


264  MECHANICS. 

water  in  the  reservoir,  than  the  height  at  whicli  the  pres^ 
sure  of  the  air  will  sustain  a  column  of  water.  In  fact,  it 
should  be  placed  a  little  lower  than  this  limit.  The  specific 
gravity  of  mercury  being  about  13.5,  the  height  of  a  column 
of  water  that  will  counterbalance  the  pressure  of  the  atmos- 
phere may  be  found  by  multiplying  the  height  of  the  baro- 
metric column  by  IS^. 

At  the  level  of  the  sea  the  average  height  of  the  baro- 
metric column  is  2^  feet ;  hence,  the  theoretical  height  to 
which  water  can  be  raised  by  the  principle  of  suction 
alone,  is  a  little  less  than  34  feet. 

The  water  having  passed  through  the  piston-valve,  may 
be  raised  to  any  height  by  the  lifting  principle,  the  only 
limitation  being  the  strength  of  the  pump. 

There  are  certain  relations  that  must  exist  between  the 
play  of  the  piston  and  its  height  above  the  water  in  the 
reservoir,  in  order  that  the  water  may  be  raised  to  the 
piston  ;  if  the  play  is  too  small,  it  will  happen  after  a  few 
strokes  of  the  piston,  that  the  air  in  the  barrel  is  not  suffi- 
ciently compressed  to  open  the  piston-valve ;  when  this 
state  of  affairs  takes  place,  the  water  ceases  to  rise. 

To  investigate  the  relation  that  should  exist  between  the 
play  and  the  height  of  the  piston  above  the  water:  Denote 
the  play  of  the  piston,  by  p,  the  distance  from  the  surface 
of  the  water  in  the  reservoir  to  the  highest  position  of  the 
piston,  by  a,  and  the  height  at  which  the  Avater  ceases  to 
rise,  by  x.  The  distance  from  the  water  in  the  pump  to 
the  highest  position  of  the  piston  will  be  «  —  x,  and  the 
distance  to  the  lowest  position  of  the  piston,  a  —p  —  x. 
Denote  the  height  at  which  the  atmospheric  pressure  sus- 
tains a  column  of  water  in  vacuum,  by  li,  and  the  weight 
of  a  column  of  water,  whose  base  is  the  cross  section  of 
the  pump,  and  altitude  is  1,  by  w  ;  then  will  tvh  denote  the 


HYDRALLIC    AND    PNEUMATIC    MACHINES.  265 

pressure  of  the  atmosphere  exerted  upward  through  the 
water  in  the  reservoir  and  pump. 

When  the  piston  is  at  its  lowest  position,  the  pressure  of 
the  confined  air  must  be  equal  to  that  of  the  external 
atmosphere;  that  is,  to  wh.  When  the  piston  is  at  its 
highest  position,  the  confined  air  will  be  rarefied,  the  vol- 
ume occupied  being  proportional  to  its  height.  Denoting 
the  pressure  of  the  rarefied  air  by  ivh\  we  shall  have,  from 
Mariotte's  law, 

wli  :  wh'  :  :  a  —  x  :  a  —p  —  x. 

a  —  p  —  X 


h'  =  h- 


a  —  X 


If  the  water  does  not  rise  when  the  piston  is  in  its  high- 
est position,  the  j^ressiire  of  the  rarefied  air,  phis  the 
weight  of  the  column  already  raised,  will  be  equal  to  the 
pressure  of  tlie  external  atmosphere ;  or, 

// ^ 'v 

wh  - — +  wx  =  wh. 

a—x 

Solving  this  with  respect  to  x,  we  have, 


_  ft  ±  V  a*  —  4:ph 


a* 
Ifj  4ph  >  a' ;    or,  P  >  -^^ 


the  value  of  x  is  imaginary,  and  there  is  no  point  at  which 
the  water  ceases  to  rise.  Hence,  the  above  inequality 
expresses  the  relation  that  must  exist,  in  order  that  the 
pump  may  be  effective.  This  condition,  expressed  in 
words,  gives  the  following  rule : 

The  play  of  the  piston  must  he  greater  than  the  square  of 
the  distance  from  the  tvater  in  the  reservoir,  to  the  highest 
position  of  the  piston,  divided  by  four  times  the  height  at 

12 


266  MECHANICS. 

which  the  atmosphere  tvill  support  a  column  of  water  in  a 
vacuum. 

Let  it  be  required  to  find  the  least  play  of  the  piston, 
when  its  highest  position  is  16  feet  above  the  water  in  the 
reservoir,  and  the  barometer  at  28  inches. 

In  this  case, 

a  =  16  a,      and  h  =  28  in.  X  ISJ  =  378  in.  =  31J  ft. 

Hence,  i?  >  fM  ft. ;     or,    p>2^  ft. 

To  find  the  quantity  of  work  required  to  make  a  double 
stroke  of  the  piston,  after  the  water  reaches  the  spout. 

In  depressing  the  piston,  no  force  is  required,  except 
that  necessary  to  overcome  inertia  and  friction.  Neglect- 
ing these  for  the  present,  the  quantity  of  work  in  the 
downward  stroke,  may  be  regarded  as  0.  In  raising  the 
piston,  its  upper  surface  is  pressed  downward,  by  the  pres- 
sure of  the  atmosphere,  wh,  plus  the  weight  of  the  column 
of  water  from  the  piston  to  the  spout;  and  it  is  pressed 
upward,  by  the  pressure  of  the  atmosphere,  transmitted 
through  the  pump,  minus  the  weight  of  a  column  of  water, 
whose  cross  section  is  that  of  the  barrel,  and  whose  alti- 
tude is  the  distance  from  the  piston,  to  the  water  in  the 
reservoir.  If  we  subtract  the  latter  from  the  former,  the 
difierence  will  be  the  downward  pressure.  This  difference 
is  equal  to  the  weight  of  a  column  of  water,  whose  base  is 
the  cross  section  of  the  barrel,  and  whose  height  is  the 
distance  of  the  spout  above  the  reservoir.  Denoting  this 
height  by  H,  the  pressure  is  equal  to  wIL  The  path 
through  which  the  pressure  is  exerted  during  the  ascent 
of  the  piston,  is  the  play  of  the  piston,  ov  p.  Denoting 
the  quantity  of  work  required,  by  Q,  we  shall  have, 

Q  =  wpH. 


HYDRAULIC    AND    PNEUMATIC    MACHINES.  267 

But  wp  is  the  weight  of  a  volume  of  water,  whose  base 
is  the  cross  section  of  the  barrel,  and  whose  altitude  is  the 
play  of  the  piston.  Hence,  Q  is  equal  to  the  quantity  of 
work  necessary  to  raise  this  volume  of  water  from  the  level 
of  the  reservoir  to  the  spout.  This  volume  is  evidently 
equal  to  that  actually  delivered  at  each  double  stroke  of  the 
piston.  Hence,  the  quantity  of  work  expended  in  pump- 
ing, with  the  sucking  and  lifting  pump,  hurtful  resistances 
being  neglected,  is  equal  to  the  quantity  of  work  necessary 
to  lift  the  amount  of  water,  actually  delivered,  from  the 
level  of  the  reservoir  to  the  spout.  In  addition  to  this,  a 
sufficient  amount  of  power  must  be  exerted  to  overcome 
hurtful  resistances.  The  disadvantage  of  this  pump,  is 
the  irregularity  with  which  the  force  acts,  being  0  in  de- 
pressing the  piston,  and  a  maximum  in  raising  it.  This  is 
an  important  objection  when  machinery  is  employed  in 
pumping;  but  it  may  be  partially  overcome,  by  using  two 
pumps,  so  arranged,  that  one  piston  ascends  as  the  other 
descends.  Another  objection  to  the  use  of  this  pump,  is 
the  irregularity  of  flow,  the  inertia  of  the  column  of  water 
having  to  be  overcome  at  each  upward  stroke. 

Sucking  and  Forcing  Pump. 

195.  This  pump  consists  of  a  barrel,  A,  with  a  sucking 
pipe,  B,  and  a  sleeping- valve,  G,  as  in  the  pump  just  dis- 
cussed. The  piston,  (7,  is  solid,  and  is  worked  up  and 
down  by  a  lever,  E,  and  a  piston-rod,  D.  At  the  bottom 
of  the  barrel,  a  pipe  leads  to  an  air-vessel,  K,  through  a 
second  sleeping-valve,  F,  which  opens  upward,  and  closes 
by  its  own  weight.  A  delivery-pipe,  H,  enters  the  air- 
vessel  at  the  top,  and  terminates  near  the  bottom. 

To  explain  the  action  of  this  pump,  suppose  the  piston, 
C,  to  be  in  its  lowest  position.     If  the  piston  be  raised  to 


268 


MECHANICS. 


P 


K 


Fig.   159. 


its  highest  position,  the  air  in  the  barrel  is  rarefied,  its  ten- 
sion is  diminished,  the  air  in  the  tube,  B,  thrusts  open  the 
valve,  and  a  portion  escapes  into  the  ^ 
barrel.  The  pressure  of  the  external 
air  then  forces  water  up  the  pipe,  B, 
until  the  tension  of  the  rarefied  air, 
plus  the  weight  of  the  water  raised, 
is  equal  to  the  tension  of  the  external 
air.  An  equilibrium  being  produced, 
the  valve,  G,  closes  by  its  own  weight. 
If  the  piston  be  depressed,  the  air  in 
the  barrel  is  condensed,  its  tension 
increases  till  it  becomes  greater  than 
that  of  the  external  air,  when  the 
valve,  F,  is  thrust  open,  and  a  portion  escapes  through  the 
delivery-pipe,  11.  After  a  few  double  strokes  of  tlie  piston, 
the  water  rises  through  the  valve,  G,  and,  as  the  piston  de- 
scends, is  forced  into  the  air-vessel,  the  air  is  condensed  in 
the  upper  part  of  the  A^essel,  and,  acting  by  its  elastic  force, 
forces  a  portion  of  the  water  up  the  delivery-pipe  and  out 
at  the  spout,  P.  The  object  of  the  air-vessel  is,  to  keep  up 
a  continued  stream  through  the  pipe,  II,  otherwise  it  would 
be  necessary  to  overcome  the  inertia  of  the  entire  column 
of  water  in  the  pipe  at  every  double  stroke.  The  flow 
having  commenced,  a  volume  of  water  is  delivered  from 
the  spout,  at  each  double  stroke,  equal  to  that  of  a  cylin- 
der whose  base  is  the  area  of  the  piston,  and  whose  alti- 
tude is  the  play  of  the  piston. 

The  same  relation  between  the  parts  should  exist  as  in 
the  sucking  and  lifting  pump. 

To  find  the  quantity  of  work  consumed  at  each  double 
stroke,  after  the  flow  has  become  regular,  hurtful  resistance 
being  neglected : 


HYDRAULIC    AND    PNEUMATIC    MACHINES.  209 

AVhen  the  piston  is  descending,  it  is  pressed  downward 
by  the  tension  of  the  air  on  its  upper  surface,  and  upward 
by  the  tension  of  the  atmosphere,  transmitted  through  the 
delivery -pipe,  j'j/?^^  the  weight  of  a  cohimn  of  w^ater  whose 
base  is  the  area  of  the  piston,  and  whose  altitude  is  the 
distance  of  the  spout  above  the  piston.  This  distance  is 
variable  during  the  stroke,  but  its  mean  value  is  the  dis- 
tance of  the  middle  of  the  play  below  the  spout.  The 
diflferenc^  between  these  pressures  is  exerted  upward,  and 
is  equal  to  the  weight  of  a  column  of  w^ater  whose  base  is 
the  area  of  tlie  piston,  and  whose  altitude  is  the  distance 
from  the"  middle  of  the  play  to  the  spout.  The  distance 
through  which  the  force  is  exerted,  is  the  play  of  the  piston. 
Denoting  the  quantity  of  work  during  the  descending 
stroke,  by^',  the  weight  of  a  column  of  water,  whose  base 
is  the  area  of  the  piston,  and  altitude  is  1,  by  w,  and  the 
height  of  the  spout  above  the  middle  of  the  play,  by  lb, 
we  have, 

Q'  =  wh'  X  p. 

When  the  piston  is  ascending,  it  is  pressed  downward 
by  the  tension  of  the  atmosphere  on  its  upper  surface,  and 
upward  by  the  tension  of  the  atmosphere,  transmitted 
through  the  water  in  the  reservoir  and  pump,  minvs  the 
weight  of  a  column  of  water  whose  base  is  the  area  of  the 
piston,  and  whose  altitude  is  the  height  of  the  piston  above 
the  reservoir.  This  height  is  variable,  but  its  mean  value 
is  the  height  of  the  middle  of  the  play  above  the  reservoir. 
The  distance  through  which  this  force  is  exerted,  is  the 
play  of  the  piston.  Denoting  the  quantity  of  work  during 
the  ascending  stroke,  by  Q",  and  the  height  of  the  middle 
of  the  play  above  the  reservoir,  hyh",  we  have, 

Q"  =  ivh"  X  i>. 


270 


MECHANICS. 


Denoting   the   entire   quantity  of  work  during  a  double 
stroke,  by  Q,  we  have, 

Q  =  q  ^  Q"  =  wp{li'  +  h"). 

But  wp  is  the  weight  of  a  volume  of  water,  whose  base  is 
the  piston,  and  wiiose  altitude  is  the  play;  that  is,  it  is  the 
weight  of  the  volume  delivered  at  each  double  stroke. 

The  quantity,  h'  +  h",  is  the  height  of  the  spout  above 
the  reservoir.  Hence,  the  work  expended,  is  equal  to  that 
required  to  raise  the  volume  delivered  from  the  level  of 
the  reservoir  to  the  spout.  To  this  must  be  added  the 
work  necessary  to  overcome  hurtful  resistances,  as  fric- 
tion, &c. 

If  li  —  h",  we  have,  Q'  =  Q";  that  is,  the  quantity  of 
work  during  the  ascending  stroke,  equal  to  that  during 
the  descending  stroke.  Hence,  the  work  of  the  motor  is 
more  nearly  uniform,  Avhen  the  middle  of  the  play  is  at 
equal  distances  from  the  reservoir  and  spout. 


Fire-Engine. 

196.  The  fire-engine  is  a  double  sucking  and  forcing 
pump,  the  piston-rods  being  so  connected,  that  when  one 
piston  ascends,  the  other  descends.  The  sucking  and  de- 
livery pipes  are  made  of  leather,  and  attached  to  the  machine 
by  metallic  screw-joints. 

The  figure  exhibits  a 
cross  section  of  the  essen- 
tial part  of  a  fire-engine. 

A,  A',  are  the  barrels, 
the  pistons  are  connect- 
ed by  rods,  i),  /),  with 
the  lever,  E,  E' ;  B  is  the 
si";king-pipe,  terminating  rig.  leo. 


HYDRAULIC  AND  PNEUMATIC  MACHINES. 


271 


in  a  box  from  whicn  the  water  may  enter  either  barrel 
through  the  valves,  G,  O ';  K  is  the  air-vessel,  common 
to  both  pumps,  and  communicating  with  them  by  valves, 
Fy  F';  H  is  the  delivery-pipe. 

It  is  mounted  on  wheels  for  convenience  of  locomotion. 
The  lever,  E,  E',  is  worked  by  rods  at  right  angles  to  the 
lever,  so  arranged  that  several  men  can  apply  their 
strength  at  once.  The  action  of  the  pump  differs  in  no 
respect  from  that  of  the  forcing  pump;  but  when  the 
instrument  is  worked  vigorously,  a  large  quantity  of  water 
is  forced  into  the  air  vessel,  the  tension  of  the  air  is  much 
augmented,  and  its  elastic  force,  thus  brought  into  play, 
propels  the  water  to  a  considerable  distance  from  the 
delivery-pipe.  It  is  this  capacity  of  throwing  a  jet  of 
water  to  a  great  distance,  that  gives  to  the  engine  its  value 
in  extinguishing  fires. 

A  pump  similar  to  the  fire-engine,  is  often  used,  under  the 
name  of  the  double-action  forcing  jJump,  for  other  purposes. 


The  Rotary  Pump. 

197.  The  rotary  pump  is  a  modification  of  the  sucking 
and  forcing  pump.  Its  construction  will  be  understood 
from  the  drawing,  which  repre- 
sents a  section  through  the  axis 
of  the  sucking-pipe,  at  right 
angles  to  the  axis  of  the  rotating 
portion. 

A  is  a  ring  of  metal,  revolv- 
ing about  an  axis;  D,  D,  is  a 
second  ring  of  metal,  concentric 
with  the  first,  and  forming  with 
it  an  intermediate  annular  space. 
This  space  communicates  with  the  sucking-pipe,  K,  and 


Fig.  161. 


272  MECHANICS. 

the  delivery-pipe,  L.  Four  radial  paddles,  C,  are  so  dis- 
posed as  to  slide  backward  and  forward  through  suitable 
openings  in  the  ring,  A,  and  are  moved  around  with  it. 
6r  is  a  guide,  fastened  to  the  end  of  the  cylinder  enclosing 
the  revolving  apparatus,  and  cut  as  represented  in  the 
figure;  E,  E,  are  springs,  attached  to  the  ring,  D,  and 
acting,  by  their  elastic  force,  to  press  the  paddles  firmly 
against  the  guide.  These  springs  do  not  impede  the  flow 
of  water  from  the  pipe,  K,  and  ifito  the  pipe,  L. 

When  the  axis,  0,  revolves,  each  paddle,  as  it  passes  the 
partition,  is  pressed  against  the  guide,  but  is  forced  out 
again,  by  the  form  of  the  guide,  against  the  wall,  D.  Each 
paddle  drives  the  air  in  front  of  it  in  the  direction  of  the 
arrow-head,  and  finally  expels  it  through  the  pipe,  L.  The 
air  behind  the  paddle  is  rarefied,  and  the  pressure  of  the  ex- 
ternal air  forces  a  column  of  water  up  the  pipe.  After  a  few 
revolutions,  the  air  is  entirely  exhausted  from  the  pipe,  K. 
The  water  enters  the  channel,  C,  C,  and  is  forced  up  the 
pipe,  L,  from  which  it  escapes  by  a  spout.  The  work 
expended  in  raising  a  volume  of  water  to  the  spout,  by 
this  pump,  is  equal  to  that  required  to  lift  it  from  the  level 
of  the  cistern  to  the  spout.  This  may  be  shown  in  the  same 
manner  as  was  explained  under  the  head  of  tlie  sucking  and 
forcing  pump.  To  this  quantity  of  w^ork,  must  be  added 
the  work  necessary  to  overcome  hurtful  resistances. 

A  machine,  similar  to  the  rotary  pump,  is  constructed 
for  exhausting  foul  air  from  a  mine;  or,  by  reversing  the 
direction  of  rotation,  to  force  fresh  air  to  the  bottom  of  the 
mine. 

The   Hydrostatic   Press. 
198.  The  hydrostatic  press  is  a  machine  for  exerting  a 
great  pressure  through  a  small  space.     It  is  used  in  com- 


HYDRAULIC    AND    PNEUMATIC    MACHINES. 


273 


pressing   seeds   to  obtain  oil,  in  packing   hay  and   other 

goods,   also   in   raising  great  weights.      Its  construction, 

though   requiring  the   use  of  a 

sucking  pump,  depends  upon  the 

principle  of  equal  pressures,  (Art. 

145). 

It  consists  of  two  cylinders,  A 
and  B,  each  provided  with  a 
solid  piston.  The  cylinders  com- 
municate by  a  pipe,  C,  whose 
entrance  to  the  larger  cylinder  is  ^'g-  ^^2. 

closed  by  a  sleeping-valve,  B.  The  smaller  cylinder  com- 
municates with  the  reservoir,  JC,  by  a  sucking-pipe,  IT, 
whose  upper  extremity  is  closed  by  a  sleeping- valve,  B. 
The  piston,  B,  is  worked  by  the  lever,  G^  By  raising  and 
depressing  the  lever,  G,  the  water  is  raised  from  the  reser- 
voir and  forced  into  the  cylinder,  A  ;  and  when  the  space 
below  the  piston,  F,  is  filled,  a  force  is  exerted  upward,  as 
many  times  greater  than  that  applied  to  B,  as  the  area  of  F 
is  greater  than  B,  (Art.  145).  This  force  may  be  utilized 
in  compressing  a  body,  L,  between  the  piston  and  the 
frame  of  the  press. 

Denote  the  area  of  the  larger  piston,  by  F,  of  the  smaller, 
by  J??,  the  pressure  applied  to  B,  by /,  and  that  exerted  at 
F,  by  F;  we  shall  have, 

p  ' 

If  we  denote  the  longer  arm  of  the  lever,  G,  by  L,  tli^ 
shorter  arm,  by  I,  and  the  force  applied  at  the  extremity  of 
the  longer  arm,  by  IT,  we  have,  from  the  principle  of  the 
lever,  (Art.  64), 

'-     K:f::l:L,     .:  f  =  -^- 


F:f::P:p, 


F 


274  MECHANICS. 

Substituting  above,  we  have, 

pi    ■ 

To  illustrate,  let  the  area  of  the  larger  piston  be  100 
square  inches,  that  of  the  smaller  piston  1  square  inch,  the 
longer  arm  of  the  lever  30  inches,  the  shorter  arm  2  inches, 
and  let  a  force  of  100  pounds  be  applied  at  the  end  of  the 
longer  arm  of  the  lever;  to  find  the  pressure  on  F. 

From  the  conditions, 

P  =  100,  K=  100,  L  =  30,  jy  r=  1,  and  /  =  2. 
Hence, 

^=15^^f^ii-«  =  150,000  lbs. 

We  have  not  taken  into  account  the  hurtful  resistances, 
hence  the  pressure  of  150,000  pounds  must  be  somewhat 
diminished. 

The  volume  of  water  forced  from  the   smaller  to  the 

larger  cylinder,  during  a  single  descent  of  the  piston,  B  , 

will  occupy,  in  the  two  cylinders,  spaces  whose  heights  are 

inversely  as  the  areas  of  the  pistons.     Hence,  the  path, 

over  which  /  is  exerted,  is  to  the  path  over  which  F  is 

exerted,  as  P  is  to  p.     Or,  denoting  these  paths  by  s  and 

S,  we  have, 

s  :  S  \\  P  \  p; 

or,  since  P  :  p  :  :  F  :  f,  we  shall  have, 

s  :  S  ::  F:f,      .-.  fs  =  FS. 

That  is,  the  work  of  the  potver  and  resistance  are  equal,  a 
principle  that  holds  good  in  all  machines. 

Examples. 

1.  The  cross  section  of  a  sucking  and  forcing  pump  is  6  square 
feet,  the  play  of  the  piston  3  feet,  and  the  lieight  of  the  spout,  above 
the  reservoir,  50  feet.    What  must  be  the  eftective  horse-power  of  an 


HYDRAULIC   AND   PNEUMATIC   MACHINES.  275 

engine  to  impart  30  double  sti-okes  per  minute,  hurtful  resistances 
being  neglected  ? 

SOLUTION. 

The  number  of  units  of  work  required  to  be  performed  eacu 
minute,  is  equal  to 

6  X  3  X  50  X  62i  X  30  =  1687500  lb.  ft 
Hence, 

„  —  1687500   —  K-14  A^o 

2.  In  a  hydrostatic  press,  the  areas  of  the  pistons  are,  2  and  400 
square  inches,  and  the  arms  of  the  lever  are,  1  and  20  inches.  Re- 
quired the  pressure  on  the  larger  piston  for  each  pound  of  pressure 
on  tlie  longer  arm  of  lever?  Am.  4000  lbs. 

3.  The  areas  of  the  pistons  of  a  hydrostatic  press  are,  3  and  300 
square  inches,  and  the  shorter  ^Jmi  of  the  lever  is  one  inch.  What 
must  be  the  length  of  the  longer  ann,  that  a  force  of  1  lb.  may  pro- 
duce a  pressure  of  1000  lbs.  ?  Ans.  10  inches. 

The  Siphon. 

199.  The  siphon  is  a  bent  tube,  for  transferring  a  liquid 
from  a  higher  to  a  lower  level,  over  an  intermediate  eleva- 
tion. The  siphon  consists  of  two  branches,  J  ^ 
and  BC\oi  which  the  outer  one  is  the  longer.  To 
use  the  instrument,  the  tube  is  filled  with  the 
liquid,  the  end  of  the  longer  branch  being  stop- 
ped with  the  finger,  or  a  stop-cock ;  in  which  case, 
the  pressure  of  the  atmosphere  prevents  the  liquid 
from  escaping  at  the  other  end.  The  instrument 
is  then  inverted,  the  end,  C,  being  submerged  in  the  liquid, 
and  the  stop  removed  from  A.  The  liquid  will  flow  through 
the  tube,  and  the  flow  will  continue  till  the  level  of  the 
liquid  in  the  reservoir  reaches  the  mouth  of  the  tube,  C. 

To  find  the  velocity  with  which  water  will  issue  from 
the  siphon,  let  us  consider  an  infinitely  small  layer  at  the 
orifice,  ji.  This  layer  is  pressed  downward,  by  the  tension 
of  the  atmosphere  exerted  on  the  surface  of  the  reservoir, 
minus  the  weight  of  the  water  in  the  branch,  BD,  plus  the 


D 

i 

c 

1 

Fig 

1G4 

276  MECHANICS. 

weight  of  the  water  in  the  branch,  BA.  It  is  pressed  up- 
ward by  the  tension  of  the  atmosphere.  The  difference  of 
these  forces,  is  the  weight  of  the  water  in  DA,  and  the  velocity 
of  the  stratum  will  be  due  to  that  depth.  Denoting  the  ver- 
tical height  of  DA,  by  7i,  we  shall  have,  for  the  velocity, 


This  is  the  theoretical  velocity,  but  it  is  never  quite  real- 
ized in  practice,  on  account  of  resistances,  that  have  been 
neglected  in  the  preceding  investigation. 

The  siphon  may  be  filled  by  applying  the  mouth 
to  the  end,  A,  and  exhausting  the  air  by  suction. 
The  tension  of  the  atmosphere,  on  the  upper  sur- 
face of  the  reservoir,  presses  the  water  up  the 
tube,  and  fills  it,  after  which  the  flow  goes  on  as 
before.  Sometimes,  a  sucking- tube,  AD,  is  inserted  near 
the  opening,  A,  rising  nearly  to  the  bend  of  the  siphon. 
In  this  case,  the  opening,  ^4, is  closed,  and  the  air  exhausted 
through  the  sucking-tube,  AD,  after  which  the  flow  goes  on 
as  before. 

The  Wurtemburg  Siphon. 

200.  In  the  Wurtemburg  siphon,  the  ends  of  the  tube 
are  bent  twice,  at  right  angles,  as  shown  in  the  figure.  The 
advantage  of  this  is,  that  the  tube,  once  filled,  re-  ^ 
mains  so,  as  long  as  tlie  plane  of  its  axis  is  kept 
vertical.  The  siphon  may  be  lifted  out  and  re- 
placed at  pleasure,  thereby  stopping  and  repro-  [^  jv^ 
ducing  the  flow  at  will.  ,      Fig.  i65. 

It  is  to  be  observed  that  the  siphon  is  only  efi'ective  when 
the  distance  from  the  highest  point  of  the  tube  to  the  level 
of  the  water  in  the  reservoir  is  less  than  the  height  at 
which  the  atmospheric  pressure  sustains  a  column  of  water 
in  a  vacuum.     This  will,  generally,  be  less  than  S-i  feet. 


HYDRAULIC   AND   PNEUMATIC    MACHINES.  277 

The  Intermitting  Siphon 

201.  The  intermitting  siphon  is  represented  in  the  figure. 
AB  is  a  curved  tube  issuing  from  the  bottom  of  a  reservoir. 
The  reservoir  is  supplied  with  water  by 
a  tube,  B,  having  a  smaller  bore  than 
the  siphon. 

To  explain  its  action,  suppose  the 
reservoir  to  be  empty,  and  the  tube,  E, 
to  be  open;  as  soon  as  the  reservoir 
is  filled  to  the  level,  CD,  the  water  be- 
gins to  flow  from  the  opening,  B,  and 
the  flow  once  commenced,  continues  till  the  level  of  the 
reservoir  is  reduced  to  CD',  through  the  opening,  A.  The 
flow  then  ceases  till  the  cistern  is  again  filled  to  CD,  and 
so  on  as  before. 

Intermitting  Springs. 

202.  Let  A  represent  a  subterranean  cavity,  communi- 
cating with  the  surface  of  the  earth  by  a  channel,  ABC, 
bent  like  a  siplion.  Suppose  the  reser- 
voir to  be  fed  })y  percolation  through 
the  crevices,  or  by  a  small  channel,  D. 
When  the  water  in  the  reservoir  rises 
to  the  horizontal  plane,  BD,  the  flow  ^^*»  ^^'• 
commences  at  C,  and,  if  the  channel  is  sufficiently  large, 
tlie  flow  continues  till  the  water  is  reduced  to  the  level 
plane  through  C.  An  intermission  then  occurs  till  the 
reservoir  is  again  filled;   and  so  on,  intermittingly. 

Siphon  of  Constant  Flow. 

203.  We  have  seen  that  the  velocity  of  efilux  depends  on 
the  height  of  water  in  the  reservoir  above  the  external 


278 


MECHANICS. 


opening  of  the  siphon.  When  the  water  is  drawn  from  the 
reservoir,  the  surface  sinks,  this  height  diminishes,  and, 
consequently,  the  velocity  continually  diminishes. 

If,  however,  the  shorter  branch,  CD,  be  passed  through 
a  cork  large  enough  to  float  the  siphon,  the  instrument 
will  sink  as  the  upper  surface  is  depressed,  the  height  of 
DA  will  remain  constant,  and,  consequently,  the  flow  will 
be  uniform  till  the  siphon  comes  in  contact  with  the  upper 
edge  of  the  reservoir.  By  suitably  adjusting  the  siphon  in 
the  cork,  the  velocity  of  efflux  can  be  increased  or  de- 
creased within  certain  limits.  In  this  manner,  any  desired 
quantity  of  the  fluid  can  be  drawn  off  in  a  given  time. 

The  siphon  is  used  in  the  arts,  for  decanting  liquids.  It 
is  also  employed  to  draw  a  portion  of  a  liquid  from  the 
interior  of  a  vessel  when  that  liquid  is  overlaid  by  one  of 
less  specific  gravity. 


The  Hydraulic  Ram. 

204.  The  hydraulic  ram  is  a  machine  for  raising  water  by 
means  of  shocks  caused  by  the  sudden  stoppage  of  a  stream 
of  water. 

It  consists  of  a  reservoir,  B,  supplied  by  an  inclined 
pipe.  A;  at  the  upper  surface  of  the  reservoir,  is  an  orifice 
closed  by  a  valve,  D  ;  this  valve  is 
kept  in  place  by  a  metallic  frame- 
work immediately  below  the  ori- 
fice ;  6r  is  an  air-vessel  communi- 
cating with  the  reservoir  by  an 
opening,  F,  with  a  spherical  valve, 
B;  this  valve  closes  the  orifice,  F, 
except  when    forced   upward,    in 

^  -^  '  Fig.  168. 

which  case  its  motion  is  restrained 

by  a  framework  or  cage  ;  ^  is  a  delivery-pipe  entering  the 


HYDRAULIC    AND    PNEUMATIC    MACHINES.  279 

air-vessel  at  its  upper  part,  and  terminating  near  the  bot- 
tom. At  P  is  a  small  valve,  to  supply  the  loss  of  air  in 
the  air-vessel,  arising  from  absorption. 

To  explain  the  action  of  the  instrument,  suppose  it 
empty,  and  the  parts  in  equilibrium.  If  a  current  of 
water  be  admitted  to  the  reservoir,  through  the  pipe.  A, 
the  reservoir  is  soon  fillecl,  and  the  water  commences 
rushing  out  at  D;  the  impulse  cf  tlie  water  forces  the 
valve,  />,  upward,  and  closes  the  opening;  the  velocity  of 
the  water  in  the  reservoir  is  checked ;  the  reaction  forces 
open  the  valve,  E,  and  a  portion  of  the  water  enters  the 
air-chamber,  G;  the  force  of  the  shock  having  been 
expended,  the  waives  both  fall  by  their  own  weight;  a 
second  shock  takes  place,  as  before ;  an  additional  quan- 
tity of  water  is  forced  into  the  air-vessel,  and  so  on  con- 
tinuously. As  the  water  is  forced  into  the  air-vessel,  the  air 
becomes  compressed;  and  acting  by  its  elastic  force,  urges 
a  stream  of  water  up  the  pipe,  JT.  The  shocks  occur  in 
rapid  succession,  and  thus  a  constant  stream  is  kept  up. 

To  explain  the  use  of  the  valve,  P,  it  may  be  remarked 
that  water  absorbs  more  air  under  a  greater,  than  under  a 
smaller  pressure.  Hence,  as  it  passes  through  the  air- 
chamber,  a  portion  of  the  contained  air  is  taken  up  by  the 
water  and  carried  out  through  the  pipe,  H.  But  each  time 
that  the  valve,  D,  falls,  there  is  a  tendency  to  a  vacuum  in 
the  upper  part  of  the  reservoir,  in  consequence  of  the  rush 
of  the  fluid  to  escape  through  the  opening.  The  pressure 
of  the  external  air  then  forces  the  valve,  P,  open,  a  portion 
of  air  enters,  and  is  afterward  forced  up  with  the  water  into 
the  vessel,  G,  to  keep  up  the  supply. 

The  hydraulic  ram  is  only  used  to  raise  small  quantities 
of  water,  as  for  the  supply  of  a  house,  or  garden.  Only  a 
small  fraction  of  the  fluid  that  enters  the  supply-pipe  actu- 


280 


MECHANICS. 


ally  passes  out  through  the  delivery-pipe ;  but  if  the  head 
of  water  is  pretty  large,  a  column  may  be  raised  to  a  great 
height.  Water  is  often  raised,  in  this  manner,  to  the 
highest  parts  of  lofty  buildings. 

Sometimes,  an  additional  air-vessel  is  introduced  over 
the  valve,  E,  to  deaden  the  shock  of  the  valve  in  its  play. 


Archimedes'  Screw. 
205.  This  is  a  machine  for  raising  water  through  small 
heights,  and,  in  its  simplest  form,  it  consists  of  a  tube 
wound  spirally  around  a  cylinder.  The  cylinder  is 
mounted  so  that  its  axis  is  oblique  to  the  horizon,  the 
lower  end  dipping  into  the  reservoir.  When  the  cylinder 
is  turned  on  its  axis,  the  lower  end  of  the  tube  describes  the 
circumference  of  a  circle,  whose  plane  is  perpendicular  to 
the  axis.  When  the  mouth  of  the  tube  comes  to  the  level 
of  the  axis  and  begins  to  ascend,  there  is  a  certain  quantity 
of  water  in  the  tube,  which  continues  to  occupy  the  lowest 
part  of  the  spire  ;  and,  if  the  cylinder  is  properly  inclined 
to  the  horizon,  this  flow  is  toward  the  upper  end  of  the 
tube.  At  each  revolution,  a  quantity  of  water  enters  the 
tube,  and  that  already  in  the  tube  is  raised,  higher  and 
higher,  till,  at  last,  it  flows  from  the  upper  end  of  the  tube. 


The  Chain  Pump. 

206.  The  chain  pump  is  an  instru- 
ment for  raising  water  through  small 
elevations. 

It  consists  of  an  endless  chain  pass- 
ing over  wheels,  A  and  B,  having  their 
axes  horizontal,  one  below  the  surface 
of  the  water,  and  the  other  above 
the    spout   of    the    pump.     Attached 


Fig.  169. 


HYDRAULIC    AND    PNEUMATIC    MACHINES. 


281 


to  tins  chain,  and  at  right  angles  to  it,  are  circular  disks, 
fitting  the  tube,  CD.  If  the  cylinder.  A,  be  turned  in  the 
direction  of  the  arrow-head,  the  buckets  or  disks  rise 
through  the  tube,  CD,  driving  the  water  before  them, 
until  it  reaches  the  spout,  C,  and  escapes.  One  great 
objection  to  this  machine  is,  the  difficulty  of  making  the 
disks  fit  the  tube.  Hence,  there  is  a  constant  leakage, 
requiring  great  additional  expenditure  of  force. 

•  Sometimes  the  body  of  the  pump  is  inclined,  in  which 
case  it  does  not  differ  much  in  principle  from  a  wheel  with 
flat  buckets,  that  has  also  been  used  for  raising  water. 


The  Air  Pump. 

207.  The  air  pump  is  a  machine  for  rarefying  air. 

It  consists  of  a  barrel.  A,  in  which  a  piston,  B,  is  worked 
up  and  down  by  a  lever,  C,  attached  to  a  piston-rod,  D. 
The  barrel  communicates  with 
a  vessel,  E,  called  a  receiver, 
by  a  narrow  pipe.  The  re- 
ceiver is  usually  of  glass, 
ground  to  fit  air-tight  on  a 
smooth  bed-plate,  KK.  The 
joint  between  the  receiver 
and  plate  may  be  rendered  more  perfectly  air-tight  by 
interposing  a  layer  of  tallow.  A  stop-cock,  H,  permits 
communication  to  be  made  at  pleasure  between  the  barrel 
and  receiver,  or  between  the  barrel  and  external  air.  When 
the  stop-cock  is  turned  in  a  particular  direction,  the  barrel 
and  receiver  communicate ;  but  on  turning  it  through  90 
degrees,  the  communication  with  the  receiver  is  cut  off, 
and  a  communication  is  opened  between  the  barrel  and 
external  air.  Instead  of  the  stop-cock,  valves  are  often 
used,  that  are  opened  and  closed  by  the  elastic  force  of  the 


Fig.  170. 


282  MECHANICS. 

air,  or  by  the  force  that  works  the  pump.  The  commu- 
nicating pipe  should  be  exceedingly  small,  and  the  piston, 
B,  when  at  its  lowest  point,  should  fit  accurately  to  the 
bottom  of  the  barrel. 

To  explain  the  action  of  the  air  pump,  suppose  the 
piston  to  be  at  its  lowest  position.  The  stop-cock,  H,  is 
turned  so  as  to  open  a  communication  between  the  barrel 
and  receiver,  and  the  piston  is  raised  to  its  highest  point 
by  a  force  applied  to  the  lever,  C.  The  air,  which  before 
occupied  the  receiver  and  pipe,  expands  so  as  to  fill  the 
barrel,  receiver,  and  pipe.  The  stop-cock  is  then  turned 
to  cut  off  communication  between  the  barrel  and  receiver, 
and  open  the  barrel  to  the  external  air,  and  the  piston 
again  depressed  to  its  lowest  position.  The  air  in  the 
barrel  is  expelled  by  the  depression  of  the  piston.  The  air 
in  the  receiver  is  now  more  rare  than  at  the  beginning,  and 
by  a  continued  repetition  of  the  process,  any  degree  of  rare- 
faction may  be  attained. 

To  measure  the  rarefaction  of  the  air  in  the  receiver,  a 
siphon-gauge  may  be  used,  or  a  glass  tube,  30  inches  long, 
may  be  made  to  communicate  at  its  upper  extremity  with 
the  receiver,  whilst  its  lower  extremity  dips  into  a  cistern 
of  mercury.  As  the  air  is  rarefied  in  the  receiver,  the  pres- 
sure on  tlie  mercury  in  the  tube  becomes  less  than  on  that 
in  the  cistern,  and  the  mercury  rises  in  the  tube.  The 
tension  of  the  air  in  the  receiver  is  indicated  by  the  differ- 
ence between  the  height  of  the  barometric  column  and  that 
of  the  mercury  in  the  tube. 

To  investigate  a  formula  for  the  tension  of  the  air  in  the 
receiver,  after  any  number  of  double  strokes,  let  us  denote 
the  capacity  of  the  receiver,  by  r,  that  of  the  connecting- 
pipe,  by  p,  and  that  of  the  space  between  the  bottom  of 
the  barrel  and  the  highest  position  of  the  piston,  by  b. 


HYDRAULIC    AND    PNEUMATIC    MACHINES.  283 

Denote  the  original  tension  of  the  air,  by  t  ;  its  tension 
after  the  first  upward  stroke  of  the  piston,  by  t' ;  after  the 
second,  third,  . .  .?^'^  upward  strokes,  by  f,  t'",  ...T'. 

The  air  which  occupied  the  receiver  and  pipe,  after  the 
first  upward  stroke,  fills  the  receiver,  pipe,  and  barrel :  ac- 
cording to  Mariotte's  law,  its  tension  in  the  two  cases 
varies  inversely  as  the  volumes  occupied ;  hence, 

t  :  t'  ::  p  +  r  +i  '.  2^  +  r,      ,'.  t'  =  ^^— -^. 

p  +  r  -\-  b 

In  like  manner,  we  shall  have,  after  the  second  upward 

stroke, 

p  +  r 


t'  :  t"  :  :  p  ■}-  r  -\-  b  :  p  +  r,       .'.  t"  —  f 


p  -\-  b  +  r 

Substituting  for  t'  its  value,  deduced  from  the  preceding 
equation,  we  have, 

\p  -{-  b  -}-  rj 
In  like  manner,  we  find, 

\p  -\-b  +  rj  ' 
and,  in  general,  after  the  n'"  stroke, 

\p  +  b  +  rJ 

If  the  pipe  is  exceedingly  small,  its  capacity  may  be  neg- 
lected in  comparison  with  that  of  the  receiver,  and  we  then 
have, 

V^  +  rJ 

Let  it  be  required,  for  example,  to  determine  the  tension 
of  the  air  after  5  upward  strokes,  when  the  capacity  of  the 
barrel  is  one-third  that  of  the  receiver. 


284  MECHANICS. 

T 

In  this  case,  ^  =  j,  and  n  =  5,  whence, 


10 

Hence,  the  tension  is  less  than  a  fourth  part  of  that  of 
the  external  air. 

Instead  of  the  receiver,  the  pipe  may  be  connected  by  a 
screw-joint  with  any  closed  vessel,  as  a  hollow  globe,  or  glass 
flask.  In  this  case,  by  reversing  the  direction  of  the  stop- 
cock, in  the  up  and  down  motion  of  the  piston,  the  instru- 
ment may  be  used  as  a  condenser.  When  so  used,  the 
tension,  after  n  downward  strokes  of  the  piston,  is  giveiv 
by  the  formula. 

Taking  the  same  case  as  that  before  considered,  with  the 
exception  that  the  instrument  is  used  as  a  condenser 
instead  of  a  rarefier,  we  have,  after  5  downward  strokes, 

r  =:  it. 

That  is,  the  tension  is  eight-thirds  that  of  the  external  air. 
external  air. 

Artificial  Fountains. 

208.  An  artificial  fountain  is  an  instrument  by  which  a 
liquid  is  forced  upward  in  the  form  of  a  jet,  by  the  tension 
of  condensed  air.  The  simplest  form  of  artificial  fountain 
is  called  Hero's  ball. 

Hero's  Ball. 

209.  This  consists  of  a  globe.  A,  into  the  top 
of  which  is  inserted  a  tube,  B,  reaching  nearly 
to  the  bottom  of  the  globe.  This  tube  is  pro- 
vided with  a  stop-cock,  0,  by  which  it  may  he 
(closed,  or  opened  at  pleasure.     A  second  tube,  D,      ^^^  ^^^' 


HYDRAULIC    AND    PNEUMATIC    MACHINES. 


285 


enters  the  globe  near  the  top,  which  is  also  provided 
with  a  stop-cock,  E. 

To  use  the  instrument,  close  the  stop-cock,  C,  and  fill 
the  lower  portion  of  the  globe  with  water  through  D ; 
then  connect  I)  with  a  condenser,  and  pump  air  into  the 
upper  part  of  the  globe,  and  confine  it  there  by  closing 
the  stop-cock,  E.  If,  now,  the  stop-cock,  (7,  be  opened, 
the  pressure  of  the  confined  air  on  the  surface  of  the 
water  in  the  globe  forces  a  jet  through  the  tube,  B.  This 
jet  rises  to  a  greater  or  less  height,  according  to  the  greater 
or  less  quantity  of  air  that  was  forced  into  the  globe.  The 
water  will  continue  to  flow  through  the  tube  as  long  as  the 
tension  of  the  confined  air  is  greater  than  that  of  the 
external  atmosphere,  or  till  the  level  of  the  water  in  the 
globe  reaches  the  lower  end  of  the  tube. 

Instead  of  using  the  condenser,  air  may  be  introduced 
by  blowing  with  the  mouth  through  the  tube,  D,  and  con- 
fined by  turning  the  stop-cock,  E. 

The  principle  of  Hero's  ball  is  the  same  as  that  of  the 
air-chamber  in  the  forcing-pump  and  fire-engine,  already 
explained. 


Hero's  Fountain. 

210.  Hero's  fountain  is  constructed  on  the 
same  principle  as  Hero's  ball,  except  that  the 
compression  of  the  air  is  effected  by  the  weight 
of  a  column  of  water,  instead  of  by  a  condenser. 

J  is  a  cistern,  similar  to  Hero's  ball,  with 
a  tube,  B,  extending  nearly  to  the  bottom  of 
the  cistern.  C  is  a  second  cistern  placed  at 
some  distance  below  A.  This  cistern  is  con- 
nected with  a  basin,  />,  by  a  bent  tube,  E,  and 
also  witli  the  upper  part  of  the  cistern.  A,  by 


g.  172. 


286  MECHANICS. 

a  tube,  F.  When  the  fountain  is  to  be  used,  A  is  nearly 
filled  with  water,  C  being  empty.  A  quantity  of  water  is 
then  poured  into  the  basin,  D,  which,  acting  by  its  weight, 
sinks  into  C,  compressing  the  air  in  the  upper  portion  of 
it  into  a  smaller  space,  thus  increasing  its  tension.  This 
increase  of  tension  acting  on  the  surface  of  the  water  in  A, 
forces  a  jet  through  B,  which  rises  to  a  greater  or  less 
height  according  to  tlie  greater  or  less  tension.  The  flow 
will  continue  till  the  level  of  the  water  in  A  reaches  the 
bottom  of  the  tube,  B.  The  measure  of  the  compressing 
force  on  a  unit  of  surface  of  the  water  in  C\  is  the  weight 
of  a  column  of  water,  whose  base  is  that  unit,  and  whose 
altitude  is  the  difference  of  level  between  the  water  in  D 
and  in  C. 

If  Hero's  ball  be  partially  filled  with  water  and  placed 
under  the  receiver  of  an  air-pump,  the  water  will  be  ob- 
served to  rise  m  the  tube,  forming  a  fountain,  as  the  air  in 
the  receiver  is  exhausted.  The  principle  is  the  same  as 
before;  the  flow  is  due  to  an  excess  of  pressure  on  the 
water  within  the  globe  over  that  without.  In  both  cases, 
the  flow  is  resisted  by  the  tension  of  the  air  without,  and 
is  urged  on  by  the  tension  within. 

Wine-Taster  and  Dropping-Bottle. 
211.  The  wine-taster  is  used  to  bring  up  a  small  portion 
of  wine  or  other  liquid  from  a  cask.     It  consists  of  a  tube, 
open  at  the  top,  and  terminating  below  in  a  nar-      y 
row  tube,  also  open.     When  it  is  to  be  used,  it  is 
inserted  to  any  depth  in  the  liquid,  which  rises  in 
the  tube  to  the  level  of  the  liquid  without.     The 
finger  is  then  placed  so  as  to  close  the  upper  end 
of  the  tube,  and  the  instrument  raised  out  of  the 
cask.     The  fluid  escapes  from  the  lower  end,  until     "' 


HYDRAULIC    AND    PNEUMATIC    MACHIiN^ES.  287 

the  pressure  of  the  rarefied  air  in  the  tube,  jf^Z^^s  the  weight 
of  a  column  of  liquid,  whose  cross  section  is  that  of  the 
tube,  and  whose  altitude  is  that  of  the  fluid  retained,  is 
equal  to  the  pressure  of  the  external  air.  If  the  tube  be 
placed  over  a  tumbler,  and  the  finger  removed  from  the 
upper  orifice,  the  fluid  brought  up  flows  into  the  tum- 
bler. 

If  the  lower  orifice  is  very  small,  a  few  droj^s  may  be 
allowed  to  escape,  by  taking  off  the  finger  and  immediately 
replacing  it.  The  instrument  then  constitutes  the  drop- 
ping-bottle. 

The  Atmospheric  Inkstand. 

212.  The  atmospheric  inkstand  consists  of  a  cylinder,  A, 
which  communicates  by  a  tube  with  a  second  cylinder,  B. 
A  piston,  0,  is  moved  up  and  down  in  A,  by 
means  of  a  screw,  D.     Suppose  the  spaces,  A 
and  B,  to  be  filled  with  ink.    If  the  piston, 
C,  be  raised,  the  pressure  of  the  external  air 
forces  the  ink  to  follow  it,  and  the  part,  B,  is 
emptied.     If  the  operation  be  reversed,  and         Fig.174. 
the  piston,  C,  depressed,  the  ink  is  again  'forced  into  the 
space,  B.     This  operation  may  be  repeated  at  nleasure. 

Prime  Movers. 

Definition  of  a  Prime  Mover. 

213.  A  PRIME  MOVER  is  a  contrivance,  by  means  of  which 
tthe  power  furnished  by  a  motor  is  made  to  impart  motion 
to  a  train  of  mechanism.  The  principal  motors  are,  water- 
power,  wind-power,  and  steam.  The  corresponding  prime 
movers  are,  water-wJieeU,  loindmills,  and  steam-engines^ 


288 


MECHANICS. 


Water-Wheels. 


214.  A  WATER-WHEEL  is  a  wheel  set  in  motion  by  the 
action  of  water.  Water-wheels  are  divided  into  two  classes 
— vertical  and  horizontal. 

There  are  three  principal  varieties  of  vertical  wheels : — 
oversJiot,  undershot,  and  breast  zvheels.  The  most  important 
horizontal  wheel  is  the  turbine. 

The  overshot  wheel  consists  of  a  cylindrical  drum,  A, 
terminated  at  its  ends  by  projecting 
rings,  B,  called  croivns.  The  space 
between  the  crowns  is  divided  into 
cells,  by  curved  or  angular  parti- 
tions ;  these  cells,  called  buckets,  are 
constructed  so  as  to  retain  the  water 
as  long  as  possible.  The  water  is 
delivered  by  a  sluice-wa}^  C,  near  the 
top  of  the  wheel,  and,  acting  by  its 
weight,  it  imparts  motion  to  the  wheel,  which  is  com- 
municated to  the  train  by  suitable  transmitting  pieces. 
This  wheel  is  employed  where  there  is  but  a  small  volume 
of  water,  with  considerable  fall. 

The  undershot  wheel  is  similar,  in  its  general  construc- 
tion, to  the  overshot  wheel;  the  partitions  between  the 
cells,  which  may  be  either 
plane  or  curved,  are  called 
floats.  The  water  is  de- 
livered at  the  bottom  of 
the  wheel,  and  impinging 
against  the  floats,  acts  by 
its  living  force  to  set  the 
wheel    in    motion.       The  fik-  i^e 


Fig.  r.5. 


HYDRAULIC    AND    PNEUMATIC    MACHINES. 


289 


velocity  of  the  water  depends  on  its  head,  that  is,  its  height 
in  the  reservoir,  A. 

The  breast  wheel  differs  from  the  undershot  wheel  in 
having  the  water  delivered  at  a  higher  level,  and  also  in 
having  a  casing,  or  trough, 
A,  called  a  breast,  which 
nearly  fits  the  periphery  "= 
of  the   wheel    that  revolves 


mmm. 


within  it.  In  this  wheel, 
the  water  acts  partly  by  its 
weight  and  partly  by  its 
living  force. 

The  turbine  turns  on  a 
vertical  axis,  and  its  floats  radiate  from  it,  being  curved 
somewhat  like  the  blades  of  a  screw  propeller.  The  water 
enters  at  the  centre  of  the  wheel,  flows  downward  and 
outward,  and  acts,  both  by  its  weight  and  living,  force,  to 
impart  motion  of  rotation  to  the  wheel. 


Windmills. 

215.  A  WINDMILL  is  a  wheel  set  in  motion  by  the  living 
force  of  a  current  of  air.  It  consists  of  a  horizontal  axle, 
always  parallel  to  the  direction  of  the  wind,  with  pro- 
jecting arms  carrying  sails,  set  obliquely  to  the  axis,  some- 
what like  the  blades  of  a  screw  propeller.  The  force  of 
the  wind,  which  acts  on  each  sail  at  its  centre  of  pressure, 
may  be  resolved  into  tAVo  components,  one  perpendicular, 
and  the  other  parallel  to  the  sail.  The  former  alone  is 
effective;  this  may  be  further  resolved  into  two  com- 
ponents, one  perpendicular,  and  the  other  parallel  to  the 
axis  of  rotation.  The  first  of  these  alone  is  concerned  in 
producing   rotation,  and  the  measure  of  its  effect  is  the 

13 


290  MECHANICS. 

product  of  its  intensity  by  its  lever  arm — that  is,  its  dis- 
tance from  the  axis. 

The  Steam-Engine. 

216.  A  STEAM-EN'GIN'E  is  a  contrivance  for  utilizing  the 

o 

expansive  force  of  steam.  The  term  is  generally  employed 
to  designate  not  only  the  engine  proper,  but  also  the 
various  appendages  for  generating  and  condensing  steam. 
The  relation  between  the  heat  applied  and  the  amount  of 
steam  generated,  as  also  its  general  mode  of  action,  have 
been  explained  in  a  previous  chapter. 

Varieties  of  Steam  Engines. 

217.  Steam-engines  may  be  condensing,  or  non-condens- 
ing. In  the  former,  the  steam,  after  having  acted  on  the 
piston,  is  condensed,  and  the  warm  water  returned  to  the 
boiler ;  in  the  latter,  the  steam  is  not  condensed,  but  hav- 
ing acted  on  the  piston,  is  blown  off,  into  the  air.  In  a 
condensing  engine,  steam  may  be  used  of  a  lower  tension 
than  15  lbs.  to  the  inch;  in  which  case  it  is  called  a  loio- 
pressure  engine.  In  a  non-condensing  engine,  the  steam 
must  be  of  a  greater  tension  than  15  lbs.  to  the  inch,  in 
order  that  it  may  be  blown  off  into  the  air.  An  engine  in 
which  steam  is  used  of  a  higher  tension  than  15  lbs.,  is 
called  a  liigh-pressure  engine.  A  condensing  engine  may 
be  either  high  or  lotv  pressure.  A  wo;i-condensing  engine 
must  be  high  i)ressure. 

Condensing  engines  are  more  economical  of  fuel,  but 
they  are  heavier  and  more  complex  in  construction ;  for 
this  reason  they  are  necessarily  statioiiary.  Non-condens- 
ing engines  are  used  for  locomotives ;  where  fuel  is  »ibun- 
dant  they  are  sometimes  used  as  stationary  engines. 


HYDRAULIC    AND    PNEUMATIC    MACHINES.  291 


The  Boiler  and  its  Appendages. 

218.  The  BOILER  is  a  shell  of  metal,  generally  of 
wrought  iron,  in  which  steam  is  generated.  Boilers  are  of 
various  forms.  One  of  the  simplest  is  cylindrical,  with 
hemispherical  ends.  Sometimes  two  smaller  cylinders, 
called  heaters,  are  placed  below  the  main  boiler,  and  con- 
nected with  it  by  suitable  pipes.  In  the  Cornish  boiler, 
the  cylindrical  shell  has  a  large  flue,  and  sometimes  two 
flues,  passing  through  it,  from  end  to  end.  The  tubular 
boiler  has  a  great  many  small  tubes,  or  flues,  passing 
through  it  for  the  transmission  of  flame  and  heated  gases. 
The  object  in  all  cases  is  to  generate  steam  rapidly  and 
economically.  To  accomplish  this,  the  boiler  is  set  in  the 
furnace  so  as  to  give  as  large  a  heating  surface,  in  propor- 
tion to  its  capacity,  as  possible,  and  the  flues  and  heat 
passages  are  constructed  to  keep  the  currents  of  hot  air 
and  gas  in  contact  with  the  heating  surface,  as  long  as  is 
compatible  with  free  combustion. 

The  following  are  some  of  the  principal  appendages  to 
the  boiler :  1°,  the  furnace,  or  fireplace,  with  its  flues  and 
dampers,  for  regulating  the  draft  and  keeping  up  combus- 
tion ;  2°,  the  feed  apparatus,  for  furnishing  water,  either 
from  the  condenser  or  from  a  reservoir,  to  supply  the  place 
of  that  converted  into  steam ;  3°,  the  safety-valve,  a  valve 
opening  into  the  boiler  and  secured  in  position  by  a  spring 
or  weighted  lever,  until  the  tension  of  the  steam  reaches 
the  limit  of  safety ;  4°,  the  gauge,  to  indicate  the  height  of 
the  water  in  the  boiler;  5°,  the  manometer,  for  showing  the 
actual  tension  of  the  steam  in  the  boiler ;  6°,  the  Mow-off 
apparatus,  consisting  of  a  cock  near  the  bottom  of  the 
boiler,  which,  when  opened,  permits  the  pressure  of  the 
steam  to  force  out  the  sediment  and  impurities  that  collect 


292 


MECHAl^ICS. 


there;   and,   7°,  the  steam-pipe,  that  conducts  the  steam 
from  the  boiler  to  the  engine  proper. 

The  Engine  proper. 
219.  The  essential  parts  of  the  engine  proper  are  shown 
in  the  cut.     As  the  figure  is  only  intended  to  illustrate  the 
general  principles  of  the  engine,  the  parts  are  arranged  in 
such  manner  as  to  exhibit  them  best  at  a  single  view. 


Fig.  178. 

The  cylinder  is  shown  on  the  left,  with  a  portion  broken 
away.  Its  interior  surface  is  smooth,  and  of  uniform  bore 
throughout. 


HYDRAULIC   AND    PNEUMATIC   MACHINES.  293 

The  piston,  P,  receives  the  pressure  of  the  steam,  alter- 
nately on  its  upper  and  lower  faces,  and  is  thus  made  to 
move  up  and  down  in  the  cylinder,  the  joint  between  them 
being  made  steam-tight  by  a  suitable  packing. 

The  piston-rody  A,  working  through  a  stuffing-box,  d,  and 
kept  parallel  to  the  axis  by  the  parallel  motion,  D,  D,  E, 
acts  on  one  end  of  the  working-beam,  L,  and  imparts  to  it 
an  oscillatory  motion. 

The  connecting-rod,  I,  transmits  the  oscillatory  motion 
to  the  crank,  K,  by  means  of  which  it  is  transformed  into 
rotary  motion  about  the  sliaft  of  the  engine. 

The  steam-chest,  b,  receives  the  steam  from  the  boiler 
through  the  steam-pipe,  c,  and  by  means  of  the  sliding-valve 
connected  with  the  rod,  m,  is  permitted  to  pass  through  the 
proper  channels,  or  steam-ports,  alternately  to  the  upper 
and  lower  faces  of  the  piston.  In  the  position  of  the 
engine  shown  in  the  figure,  the  steam  from  the  boiler 
passes  into  the  upper  steam-passage,  rises  to  the  top  of  the 
cylinder,  enters  it  there,  and  acts  to  force  the  piston  down; 
the  steam  below  the  piston  passes  up  through  the  lower 
steam-passage,  is  prevented  from  entering  the  steam-chest 
by  the  sliding-valve,  passes  out  at  the  opening,  a,  and  is 
thence  conveyed  by  the  eduction  pipe,  U,  to  the  condenser, 
0;  when  the  piston  reaches  the  bottom  of  the  cylinder 
the  motion  imparted  to  the  shaft  operates  on  the  eccentric, 
e,  to  move  the  eccentric  rod,  Z,  which,  in  turn,  through  the 
bent  lever,  m,  draws  the  sliding-valve  up,  so  as  to  cover  the 
upper,  and  uncover  the  lower  steam-passages  ;  the  opening, 
a,  of  the  eduction  pipe  is  then  in  communication  with  the 
upper  end  of  the  cylinder,  through  the  upper  passage  and 
the  sliding-valve.  In  this  state  of  affairs,  the  steam  from 
the  boiler  enters  the  cylinder  by  the  lower  passage,  the 
piston  is  forced  up,  the  steam  above  the  piston  is*  driven 


294  MECHANICS. 

into  the  eduction  pipe,  U,  and  thence  to  the  condenser ; 
when  the  piston  reaches  the  upper  limit  of  its  play,  the 
position  of  the  sliding-valve  is  again  reversed,  and  so  on 
continually. 

The  cold-tuater  pump,  72,  worked  by  the  rod,  H,  draws 
cold  water  from  a  reservoir,  and  forces  it  through  a  pipe, 
T,  into  the  condenser.  This  pipe,  terminating  in  a  rose, 
delivers  the  water  in  the  form  of  a  cold  shower,  which 
acts  to  condense  the  steam  that  is  continually  forced 
into  it. 

The  air-pump,  M,  worked  by  a  rod,  F,  draws  the  hot 
water,  and  the  air  that  is  mixed  with  it,  from  the  con- 
denser, and  forces  it  into  the  hot-ioell,  N. 

The  feed-pump,  Q,  worked  by  the  rod,  G,  draws  the  water 
from  the  hot- well  and  forces  it  back  to  the  boiler. 

The  Locomotive. 

220.  The  Locomotive  is  represented  in  section  by  the 
accompanying  figure.  The  essential  parts  are  the  fol- 
lowing : 

The  holler,  B,  B,  with  \i^  flues,  p,  p),  and  its  safety-valve, 
M.  The  dotted  line  shows  the  height  of  the  water  in  the 
boiler. 

The  fire-hox,  A,  communicating  with  the  smohe-hox,  C, 
by  means  of  the  flues,  p,  p.  The  fire-box  has  a  double 
wall,  the  interval  being  filled  with  water,  communicating 
with  that  in  the  boiler.  Fuel  is  supplied  by  the  door,  D, 
and  air  enters  the  fire-box  from  beloAV,  through  the  grate,  E, 

The  steam-dome,  B,  is  an  elevated  portion  of  the  boiler, 
whose  object  is  to  permit  steam  to  enter  the  steam-pipe, 
without  any  admixture  of  water,  as  might  happen  if  it 
were  taken  from  a  lower  level. 

The  steam-pipe,  S,  S,  conveys   steam  from  the  dome  to 


HYDRAULIC   AND    PKEUMATIC   MACHINES.  295 


Fig.  179. 


296  MECHANICS. 

the  steam-chest,  where  it  is  distributed  in  the  manner 
described  in  the  last  article. 

The  cylinder,  the  piston,  P,  and  the  piston-rod,  R,  are 
similar  to  the  corresponding  parts  of  the  engine  described 
in  the  last  article. 

The  steam,  after  acting  on  the  piston,  is  blown  off 
through  the  blast-pipe,  L,  which  terminates  in  the  smoke- 
box,  C.  The  current  produced  increases  the  draft,  and 
thus  promotes  the  combustion  of  the  fuel. 

The  connecting-rod,  G,  transmits  the  motion  of  the 
piston  to  the  crank,  which  converts  it  into  rotary  motion 
about  the  axis  of  the  driving-wheels,  F. 

The  alternate  back  and  forth  motion  of  the  sliding- 
valve  is  effected  by  an  eccentric,  placed  on  the  axle  of  the 
driving-wheel.  The  supply  of  water  is  obtained  by  pumps 
placed  under  the  frame  and  worked  by  eccentrics.  These 
suck  the  water  from  a  reservoir,  mounted  on  the  tender, 
which  is  a  car  attached  to  the  locomotive  for  the  trans- 
portation of  water  and  fuel. 


985133   ^  t-  ?6  i 


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